Physik Department. Matrix Product Formalism


 Oliver Barrett
 3 years ago
 Views:
Transcription
1 Physik Department Matrix Product Formalism Diplomarbeit von María Gracia Eckholt Perotti Angefertigt an der Technische Universität München und am MaxPlanckInstitut für Quantenoptik Garching, September 2005
2 First Adviser: Prof. Dr. Ignacio Cirac Second Adviser: Dr. Juan José García Ripoll Dr. Michael Wolf
3 Each path is only one of a million paths. Therefore, you must always keep in mind that a path is only a path. If you feel that you must not follow it, you need not stay with it under any circumstances. Any path is only a path. There is no affront to yourself or others in dropping it if that is what your heart tells you to do. But your decision to keep on the path or to leave it must be free of fear and ambition. I warn you: look at every path closely and deliberately. Try it as many times as you think necessary. Then ask yourself and yourself alone one question. It is this: Does this path have a heart? All paths are the same. They lead nowhere. They are paths going through the brush or into the brush or under the brush. Does this path have a heart is the only question. If it does, then the path is good. If it doesn t, then it is of no use. Carlos Castaneda i
4
5 Contents 1 Introduction and Motivation 1 2 Entanglement Separability Entanglement Measures Entanglement Detection Positive Maps Entanglement Witnesses Matrix Product States The Basics MPS in the AKLT model MPS from slightly entangled states MPS in DMRG Simple examples of MPS Formal Aspects Expectation values Normal Form Transfer Matrix and Normal Form Some physical properties Decay of correlation functions Entanglement An application: Calculation of groundstates DMRG in brief The Algorithm Normalization Effective Hamiltonian Writing the code Some results Matrix Product Density Operators MPDO from MPS Formal Aspects Expectation values Normal Form iii
6 iv 4.3 Entanglement in MPDO Conclusions and Outlook 47 A Notation and review of selected topics in Linear Algebra 49 A.1 Basic Mathematical Notation A.2 Selected topics in Linear Algebra B Normal Form for MPS from the Schmidt decomposition 51 C Normal Form implies normalized states for OBC 61
7 Chapter 1 Introduction and Motivation Entanglement is one of the most striking features of quantum theory. After playing a significant role in the foundations of quantum mechanics, it has been recently rediscovered as a new physical resource with potential commercial applications such as quantum cryptography, better frequency standards or quantumenhanced positioning, and clock synchronization. The ability to generate entangled states is one of the basic requirements for building quantum computers. Hence, efficient experimental methods for detection, verification, and estimation of quantum entanglement are of great practical importance. For this task we need a complete theoretical framework which provides us with the tools for characterizing entanglement. It is well known that it is not difficult to say when a quantum pure state is entangled. On the other hand, it is usually very hard to identify whether a given mixed state is entangled. Indeed, it is more relevant for experimental applications to investigate the structure of mixed state entanglement, because in real settings we often have either incomplete information about the system or interactions with the environment, known as decoherence, both of which give rise to mixed states. The ultimate aim of this work is the characterization and detection of entanglement in mixed states. We will focus on the matrixproductdensityoperator (MPDO) representation for mixed states and deal with the characterization of entanglement by means of the partialtranspose (PT): one of the standard criterion for detecting entanglement. The reason of using these specifications lies on the fact that this criterion can be implemented efficiently to mixedquantumstates written in the MPDO representation, i.e. the PT can be easily handled with only a linear growth of computational effort on the size of the system. The key to this lies in the fact that having ρ written as an MPDO, it is only determined by a set of matrices {M}. Indeed, there is not only one set but a infinite number of them (Fig. 1.1) leading to the same state ρ. In view of this, in this thesis we are interested in answering these two questions: Does there exist a standard set {M} that completely represents the class of sets that define ρ? In other words, we are looking for a set {M} in onetoone 1
8 2 Chapter 1. Introduction and Motivation Figure 1.1: All these sets define the same ρ. correspondence with the state ρ. In order to do this, we will review the normal form in the matrixproduct formalism for pure states and extend this to a normal form for mixed states. Is it possible to study entanglement properties of ρ by means of studying properties of the set {M}? In particular, we would like to see if the positivity of a density operator ρ is a necessary and sufficient condition for the positivity of this set of standard matrices describing the state in the MPDO formalism. If this is the case, it would be sufficient to check the positivity of these matrices after PT for detecting entanglement; simpler than working directly with ρ. The structure of this thesis is as follows: In chapter 2 we will consider the PT as a mathematical tool for entanglement detection, together with an overview of the theory of entanglement. Although its roots date back from the end of the 80s and considerable advances have been achieved during the last couple of years, there exists no comprehensive publication summarizing the basic features of the matrixproduct formalism. The following two chapters give a complete, selfcontained description. In chapter 3 we start introducing the matrix product states (MPS) from different perspectives: The class of MPS first saw light as the ground state of the AKLT model, an exactly solvable model in condensed matter physics. In this section we briefly explain the model and how it leads to the matrixproduct representation. MPS appear as a natural and efficient way to represent states of slightly entangled systems. We also explain briefly the roll that MPS have played in the Density Matrix Renormalization Group (DMRG) method. The chapter follows with the explicit derivation of some formal aspects, namely the calculation of expectation values using transfer matrices and the normal form for MPS. For
9 completeness, the normal form is derived from a set of conditions imposed on a transfer matrix. Later, to give a flavor of what this states represent, some physical properties are derived. To close the chapter we apply MPS to the ground state calculation within the framework of the DMRG method. We summarize the most relevant features of this technique and give a detailed account of the algorithm using MPS. Finally, we give some results for the specific problem of the Ising model. In chapter 4 we follow more or less the same structure. Here matrix product density operators are introduced as an extension of matrix product states to the mixed state scenario. We derive the calculation of expectation values using transfer matrices and work on the normal form for MPDO. To close the chapter, we study the implementation of PT on MPDO. Finally in chapter 5 we briefly summarize the results and give an outlook on future work. There are also three appendices at the end of this work. In appendix A we explain the basic mathematical notation that we use. We also include some selected topics in linear algebra to make this work more selfcontained, namely the definition of rank, unitary matrices, a brief comment on the determination of eigenvectors and eigenvalues and the singular value decomposition. In appendix B we treat the derivation of the normal form for MPS from the Schmidt decomposition used in chapter 3. Finally, in appendix C, we show that the normal form for MPS, in the case of OBC, implies normalized states. This fact simplifies in a very convenient way the formulation of the algorithm for the groundstate calculation, described at the end of chapter 3. 3
10 4 Chapter 1. Introduction and Motivation
11 Chapter 2 Entanglement If two systems interacted in the past it is, in general, not possible to assign a single state vector to either of the two subsystems [14]. This is also known as the principle of nonseparability and expresses much of what entanglement is about. First recognized by Einstein, Podolsky and Rosen [12] and Schrödinger [31], it is one of the most astonishing features of the quantum formalism. The main problem in Entanglement Theory is that we do not fully understand what entanglement is. More precisely, we only know is its mathematical definition and its manifestations [5, 7, 4]. Entanglement appears as the consequence of the combination of two of the quantum postulates: the state of a quantum system is described by a vector in a complex Hilbert space + the Hilbert space of a composite system is the tensor product of the two local spaces = superposition of pure states that cannot be written as the tensor product of pure states in each local space Antipodean to entangled states are the separable states, i.e., a state is entangled if and only if it is not separable. Whether a given state is entangled or just classically correlated is easy to determine for pure states. However, for arbitrary mixed states it is a hard problem [16]. We will see this later. 2.1 Separability Deciding whether several systems are entangled or whether they are just classically correlated is known as the separability problem. In this section we present the separability condition for pure and mixed states, i.e., the definition of entangled states. We will be referring to bipartite systems in a Hilbert space H = H A H B. 5
12 6 Chapter 2. Entanglement Pure States A pure state ψ is entangled if and only if it is not separable, i.e., it cannot be written as a product vector ψ = ψ A ψ B. In this case the criterion, for deciding if the state is entangled or not, is very simple. First we introduce an useful tool [24, 13]. Theorem 1 (Schmidt decomposition) Suppose ψ is a pure state of a composite system, AB. Then there are orthonormal states { i A } for system A, and orthonormal states { i B } of system B such that ψ = i λ i i A i B, where λ i are nonnegative real numbers satisfying i λ2 i = 1 known as Schmidt coefficients. If there is no degeneracy, this decomposition is unique up to arbitrary opposite phases in i A and i B. The Schmidt rank is defined as the number of nonvanishing Schmidt coefficients. Then, the criterion for pure states is ψ is pure ψ has Schmidt rank one. Mixed States A mixed state ρ is entangled if and only if it is not separable, i.e., it cannot be written as [39] N ] ρ = p i [ ψa ψ i A i ψb ψ i B i i=1 where N N + is arbitrary; ψ i A H A, ψ i B H B are arbitrary but normalized and p i 0 with N i=1 p i = 1. That is, a separable state can be prepared by two distant observers who receive instructions from a common classical source and prepare the different pure states ψ i A and ψi B with probability p i (Fig. 2.1). So, entangled states are those that cannot be created using local operations and classical communication. The criteria for entanglement of mixed state are many and diverse. Here we start introducing two of them [26, 20]. The symbol i indicates the transposition of subsystem i, i.e., partial transposition of the entire system with respect to i (see section 2.3) Theorem 2 (Peres) If ρ is separable then ρ A 0 and ρ B = (ρ A ) 0. Theorem 3 (Horodecki) A state ρ of a C 2 C 2 or C 2 C 3 system is separable if and only if its partial transposition is a positive operator.
13 2.2. Entanglement Measures 7 Figure 2.1: Separablestates factory. A classical source gives with probability p i the output i, indicating far away partners which state to prepare. 2.2 Entanglement Measures Quantifying quantum entanglement is one of the central topics in quantum information theory. How can entanglement be measured or quantified, how can entanglement be classified, i.e., what physically different types of entanglement exist, and finally how does entanglement behave as a physical resource for quantum communication, quantum computation, etc.? First of all, we need to know what an entanglement measure is [28]. We answer this important question by stating the conditions that every measure of entanglement E has to satisfy: Entanglement is nonnegative. It is zero if and only if the state is separable E(ψ) 0 ψ, E(ψ) = 0 ψ is separable Entanglement of independent systems is additive E(ψ n ) = ne(ψ) Entanglement is conserved under local unitary operations ψ Uψ, U = U A U B : E(ψ) = E(Uψ) a local change of basis has no effect on E Its expectation value cannot be increased by local nonunitary operations ψ local nonunitary {p j, ψ j } : p j E(ψ j ) E(ψ) monotonicity under local operations and classical communication (LOCC) For more on this see the pioneering paper on entanglement measures [8]. A pure state s entanglement is measured by its entropy of entanglement E(ψ) j ψ = i p i ψ i A ψ i B : E(ψ) = S(ρ A ) = S(ρ B ) (2.1)
14 8 Chapter 2. Entanglement i.e., the apparent entropy of any of the systems considered alone, where S(ρ) = T r(ρ logρ) (2.2) is the von Neumann entropy, ρ A = T r B ψ ψ is the reduced density matrix of A, obtained after tracing over B s degrees of freedom, and the logarithm is to base two (the information is stored in qubits). The entropy measures how much uncertainty there is in the state of the physical system. For example, if ρ A and ρ B describe pure states (there is no uncertainty in the individual systems), then E(ψ) = 0 (there are no quantum correlations between them). We define an ebit as the amount of entanglement in a maximally entangled state of two qubits, for which E = 1. Another possibility is to use the rank of the Schmidt decomposition (SD) as a measure. If A is a subset of n qubits and B the rest of them, the SD of ψ with respect to the partition A : B reads ψ = χ A α=1 λ α ψ[a] α ψ[b] α The rank χ A of ρ A (the reduced density matrix for block A) is a natural measure [37] of the entanglement between the qubits in A and those in B. Therefore, a good measure to quantify the entanglement of state ψ would be the maximal value of χ A over all possible bipartite splits A : B of the n qubits, namely or the related entanglement measure E χ χ := max χ A A E χ := log 2 (χ) In the bipartite setting, E χ upper bounds the more standard measure entropy of entanglement. For mixed states we have a whole zoo of measures, there is not a unique measure of entanglement. The choice of one measure or another depends on what you need. We will see some examples in what follows. In principle, there are two approaches to quantify entanglement [8, 19]: Abstract approach A state function can be used to quantify entanglement if it satisfies the natural properties stated before as definition of a measure. 1. Von Neumann entropy S: already introduced in (2.2). 2. Relative Entropy of Entanglement E R : it is based on the idea of distance; the closer the state is to the set of separable states, the less entangled it is.
15 2.3. Entanglement Detection 9 3. Other measures: Squashed Entanglement E sq, Rényi Entropy E α, Logarithm of the Negativity E N, Concurrence C, etc. Operational approach The system is more entangled if it allows for better performance of some task impossible without entanglement. 1. Entanglement of Formation EoF : having a large number n of Bell states, we want to produce as many (highfidelity) copies ψ using LOCC, getting finally m copies, therefore ψ s E of formation is the limiting ratio n/m. 2. Distillable entanglement E D : performing the reverse process, it is the limiting ratio m/n, when having a large number m of copies of ψ and we want to distill as many Bell states using LOCC, getting finally n EPR pairs. 3. Other measures: Entanglement Cost E C, Entanglement of Assistance EoA, etc. All these measures are equivalent in certain limits, e.g. [17]. We have so many definitions not only due to the diverse interpretations, but because calculating some of them are of the Big Open Problems of QIT. 2.3 Entanglement Detection Entangled states of many qubits are needed for quantum information tasks such as measurement based quantum computation [29], error correction [15] or quantum cryptography [10], to mention only few. Thus, it is important to study, both theoretically and experimentally, multipartite entanglement and to provide efficient methods to verify if in a given experiment entanglement is really present. Although, to detect entanglement is not an easy job. Here we introduce some ideas of two formalisms that deal with entanglement detection: positive maps and entanglement witnesses Positive Maps Any admissible physical transformation of a density matrix can be specified through some operators {K i } such that, ρ ρ i = K iρk i p i : p i = T r(k i ρk i ) (2.3) {K i } are known as Kraus operators [21]. These transformations define what is called a completely positive map (CPM), κ(ρ) = i K i ρk i which fulfils the following properties:
16 10 Chapter 2. Entanglement 1. Sends positive operators into positive operators, i.e., a positive map. 2. It s also positive for composite systems, ρ 0, κ(ρ) 0 ρ AB 0, (I A κ)(ρ AB ) 0 because any physical transformation should still remain meaningful when it is just performed on a subset of the parties. Maps that are positive, but not completely positive, define unphysical operations. This property makes them useful for the detection of entanglement: Any positive map acting on a product state gives a positive operator. Therefore, the same is valid for separable states. But if acting on some ρ AB this map produces to a nonpositive operator, then one can conclude that the state is entangled. We can see this in more detail, consider we have a separable state ρ AB = N ( ) p i ρ i A ρ i B and we apply the positive, but noncompletely positive, map κ on subsystem B ρ AB ρ AB = i=1 N ( p i ρ i A κ(ρ i B) ) i=1 given that κ is a positive map, κ(ρ i B ) will also be legitimate density matrices. So, it follows that none of the eigenvalues of ρ AB is negative. This is a necessary condition for ρ AB being separable. For every entangled state there is a positive map detecting it. This is an straightforward consequence of the following theorem [20]. Theorem 4 (Horodecki) A state ρ H A H B is separable if and only if for all positive maps ε : H B H C we have (I A ε)ρ 0 This translates the problem of detecting entanglement to the characterization of all the positive maps. We now move to the study of a concrete positive map.
17 2.3. Entanglement Detection 11 Partial Transposition The most known positive map which is not completely positive is the matrix transposition. The transposition θ is the map θ(c) C where C is the matrix obtained by exchanging C s rows and columns and it satisfies the identity (C ) 1 = (C 1 ) writing the matrix elements From this we define the map (c) i,j = c j,i. (θ I) (C) C A which is called partial transposition. If we work with the matrix elements we would write (c) A i A i B,j A j B = c ja i B,i A j B. As we saw before, right after the definition of a completely positive map, if we apply (θ I) (ρ) = ρ and ρ is a nonpositive operator, then ρ is entangled. Our criterion reduces to transpose part of ρ and diagonalize the resultant matrix. This is an easily computable criterion for entanglement in mixed states [26], seen in theorem 2. A drawback of PT is that it is not a sufficient condition; it has only been proved to be a sufficient condition for pure states and for composite systems having dimensions 2 2 and 2 3 [20]. Regardless of the fact that PT is a nonphysical operation (consequently, it cannot be use to detect entanglement experimentally), it can be understood as antiunitary time inversion operation in one subsystem; it means that e.g. Alice inverses time while Bob does not. We can understand more this and the effectiveness of PT in the following. According to Wigner s theorem [41], every symmetry transformation should always be implemented by a unitary (U) or antiunitary (A) matrix. If we are working with a binary composite system, i.e., H = H a H b, the direct product of unitary matrices U a U b (or antiunitary matrices A a A b ) is a unitary (or antiunitary) matrix in H. Nevertheless, the combination of a unitary and an antiunitary transformation U a A b (or A a U b ) results in a transformation which is neither unitary nor antiunitary in H, whose action on a general ket of the composite system ψ H, furthermore, cannot be properly defined [30]. However, its action on a product state is, but for a phase ambiguity, well defined. As a separable state ρ s H can always be rewritten as a statistical mixture of product states, ρ s = p i ( a i a i b i b i ), 1 p i 0, p i = 1 i i the action of such operations on ρ s leads to a ρ s ρ s ρ s = i p i ( a i a i b i b i )
18 12 Chapter 2. Entanglement where a i := U a a i H a, b i := A b b i H b, which is also physical (a positive defined hermitian matrix with normalized trace). Separable states are characterized by this: any local 1 symmetry transformation, which obviously transforms local physical states into local physical states, also transforms the global physical state into another physical state. There is only one independent antiunitary symmetry which physical meaning is time reversal. Any other antiunitary transformation can be expressed as the product of a unitary matrix times time reversal. Therefore, quantum separability of composite systems implies the lack of correlation between the time arrows of their subsystems, as if separable systems do not have memory of a unique time direction in the sense entangled states have and they are thus compatible with a time evolution which factorizes into the product of two opposed time evolutions still leading to a physical state Entanglement Witnesses Entanglement detection in an experiment is a hard problem, since reconstructing the whole density matrix is usually not possible and the quantum state is only partially known. One can typically measure a few observables and still one would like to detect some of the entangled states. In this direction appears another approach for detecting entanglement, the socalled entanglement witnesses (EW). An EW is a hermitian operator (an observable) W such that if T r[w ρ] < 0, then ρ is entangled. For every entangled state there is an EW W detecting it. This is a consequence of a special formulation of the HahnBanach theorem: Theorem 5 (HahnBanach) Let S be a convex set in a finite dimensional Banach space. Let ρ be a point in the space with ρ / S. Then there exists a hyperplane 2 that separates ρ from S. Therefore, for every entangled ρ / S 3 there exists a hyperplane, described by a Hermitian operator W 4, which separates ρ from S, such that T r[ρw ] < 0, whereas σ S : T r[σw ] 0. See Fig. (2.2) As theorem 4 for positive maps, this theorem is quite powerful from a theoretical point of view. However, we have the same handicap: it is not useful for constructing witnesses that detect entanglement in a given state ρ. An entanglement witness only gives one condition at detecting entanglement, while for a map has to be positively definite (i.e., there are many that have to be fulfilled). Thus 1 Local means that it refers to the subsystem 2 A linear subspace with dimension one less than the dimension of the space itself. 3 S, set of separable ρ s, is a convex and closed set. 4 In operator space, Hermitian operators define planes: {ρ : T r[ρw ] = const.}.
19 2.3. Entanglement Detection 13 Figure 2.2: Geometric HahnBanach theorem. W is a valid entanglement witness for all entangled ρ s in the redlined zone. a map is much stronger. However, EW are able to provide a more detailed classification of entangled states [32].
20 14 Chapter 2. Entanglement
21 Chapter 3 Matrix Product States A positiondependent unnormalized matrix product state for a onedimensional system of size N is defined as, ψ mps = d s 1,...s N =1 T r (A[1] s 1 A[2] s 2...A[N] s N ) s 1,...s N (3.1) A[i] s i : Matrix associated to site i and its state s i, whose dimension is bounded by some fixed number D i D i+1. They parametrize the state. d : Dimension of the Hilbert space corresponding to the physical system. They are a class of states that yields local descriptions of multipartite quantum states, giving a very good approximation [33] with only a polynomial number of parameters in some 1D problems. In the special case of open boundary conditions (OBC) we have D 1 = D N+1 = 1. In this chapter we start introducing matrix product states from three different perspectives. First, from their roots as the ground state of the AKLT model, where they were originally introduced as ValenceBond Solid (VBS) states. Second, in a more mathematical scenario through the Schmidt decomposition. Third, we give a short description of their roll in the Density Matrix Renormalization Group (DMRG). We end this first introductory section with some examples. Later, we treat formal aspects of the formalism concerning calculations of expectation values and the definition of the matrices A for a given state. Then, we study some physical properties of these states. To close this chapter, we deal with an application of all the machinery we have built: the calculation of ground states and correlation functions. 3.1 The Basics We acquaint the reader three different pictures of matrix product states. We start with a physical approach, follow later in a more mathematical direction and end with 15
22 16 Chapter 3. Matrix Product States their appearance within a numerical method that during the last years has become very successful simulating condensedmatter systems. The purpose of this section is merely to give an idea of where do the MPS come from MPS in the AKLT model The AKLT [2, 3] is an exactly solvable model of an antiferromagnetic spin1 chain exhibiting strong quantum fluctuations. Proposed by Affleck, Kennedy, Lieb and Tasaki, they were able to establish the existence of a Haldane gap (and related phenomena) rigorously. This was the first rigorous demonstration that exotic behavior in integer spin chains predicted by Haldane is indeed possible. We will briefly summarize it here. The goal was to create an instance for a Hamiltonian, for a onedimensional isotropic spin chain, with a continuous symmetry, exponentially decaying correlation functions, a gap and a unique (in the thermodynamic limit) ground state. Figure 3.1: A Valence Bond. The key to the model is the idea of a valence bond. Given two spin1/2 s, a valence bond is formed by putting them in the singlet state, as represented in Fig As depicted in Fig. 3.2, consider a spin1 chain. Each spin can be regarded as the symmetric part of the product of two virtual spin1/2 s = 0 1 Antisym. Symmetric We construct a state with a valence bond between each pair of adjacent sites i and i + 1 by forming a singlet out of one of the spin1/2 s at site i and one at site i + 1 (this also preserves the translational invariance in a simple way) [1]. After doing this we must symmetrize the two spin1/2 s at each site to restore a spin1 at each site. is If we label the virtual spins at each site α and β, the state of the spin at a given site
23 3.1. The Basics 17 Figure 3.2: Valence Bond Solid state (VBS). s αβ := 1 2 ( α β + β α ) for α, β =,, while s αβ = s βα denote the same state. This gives us an orthogonal basis in the symmetric part of the tensor product (LHS refers to the real space, RHS refers to the virtual space): +1 = s 2 0 = s = s 1 = s 2 or equivalently, we can apply the following projector over each site to map the state to the real space P = +1 s + 0 s + 0 s + 1 s Generalizing to different projectors P [i] for different sites i: P [i] = s i,α,β A[i] s i α,β s i αβ (3.2) where the elements of the matrix A denote the coefficients in the projector P. For a system of two particles we can make the following construction
24 18 Chapter 3. Matrix Product States AKLT [N=2] α1 β 2 = β 1,α 2 ε β 1α 2 s α1 β 1 s α2 β 2 (3.3) where ε is the LevyCivita antisymmetric tensor, it preserves the rotational invariance and contracts two adjacent virtual spins from different sites to form a singlet. This kind of state has only spin 0 or 1. We construct the Hamiltonian H as the sum of link Hamiltonians H i for each site. If we want to have (3.3) as ground state, we choose H as the sum of projection operators onto spin2 for each neighboring pair H = H i = [ 1 S 2 i S i ( S i S i+1 ) ] (3.4) 3 i i where S i are spin1 operators. Obviously H 0, since H i are projectors. Therefore, if we find an state such that H i s = 0 for all i, it will be the ground state. Thus, (3.3) is effectively the exact ground state for (3.4) with groundstate energy zero. Considering a chain of arbitrary size N, this ground state can be written as the sum of tensor products of states s 1 s n with the coefficients c s1...s N expressed as products of matrices A[1] s 1... A[N] s N, i.e., a matrixproduct state. This can be easily seen when we map explicitly (3.3) to the real space using (3.2) s at each site for generic coefficients A MPS from slightly entangled states Another approach to MPS is [37] by G. Vidal, developed independently to previous works. There, a particular decomposition for the coefficients of an arbitrary state is introduced ψ H2 n in the computational basis { 0, 1 } c i1 i 2 i n = ψ = 1 i 1 =1 1 c i1 i n i 1 i n i n=0 α 1,α n 1 Γ[1] i1 α1 λ[1] α1 Γ[2] i 2 α1 α 2 λ[2] α2 Γ[3] i 3 α2 α 3 Γ[n] in α n 1 (3.5) It is obvious that by contracting Λ λ s or λ Λ s together we obtain (3.1). The motivation of [37] is to show that any quantum computation with pure states can be efficiently simulated with a classical computer provided the amount of entanglement involved is sufficiently restricted. We will only describe the state decomposition and finish with a brief justification of the work. Decomposition (3.5) employs n tensors {Γ[1],..., Γ[n]} and n 1 vectors {λ[1],..., λ[n 1]}, whose indices i l and α l take values in {0, 1} and {1,..., χ}, respectively.
1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series
More informationMixed states and pure states
Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationH = = + H (2) And thus these elements are zero. Now we can try to do the same for time reversal. Remember the
1 INTRODUCTION 1 1. Introduction In the discussion of random matrix theory and information theory, we basically explained the statistical aspect of ensembles of random matrices, The minimal information
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationCHAPTER 12 MOLECULAR SYMMETRY
CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Javier Enciso encisomo@in.tum.de Joint Advanced Student School 009 Technische Universität München April, 009 Abstract In this paper, a gentle introduction to Quantum Computing
More information0.1 Phase Estimation Technique
Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design
More informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationHermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)
CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More information1 Spherical Kinematics
ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationBy W.E. Diewert. July, Linear programming problems are important for a number of reasons:
APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)
More informationUsing the Singular Value Decomposition
Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationarxiv:quantph/9607009v1 11 Jul 1996
Distillability of Inseparable Quantum Systems Micha l Horodecki Department of Mathematics and Physics University of Gdańsk, 80 952 Gdańsk, Poland arxiv:quantph/9607009v1 11 Jul 1996 Pawe l Horodecki Faculty
More informationLecture 2: Essential quantum mechanics
Department of Physical Sciences, University of Helsinki http://theory.physics.helsinki.fi/ kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics JaniPetri Martikainen
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture  17 ShannonFanoElias Coding and Introduction to Arithmetic Coding
More informationQuick Reference Guide to Linear Algebra in Quantum Mechanics
Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationBOX. The density operator or density matrix for the ensemble or mixture of states with probabilities is given by
2.4 Density operator/matrix Ensemble of pure states gives a mixed state BOX The density operator or density matrix for the ensemble or mixture of states with probabilities is given by Note: Once mixed,
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationFor the case of an Ndimensional spinor the vector α is associated to the onedimensional . N
1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. onhermitian
More informationLinear Algebra: Matrices
B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3.
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement Primary
Shape, Space, and Measurement Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two and threedimensional shapes by demonstrating an understanding of:
More informationStructure of the Root Spaces for Simple Lie Algebras
Structure of the Root Spaces for Simple Lie Algebras I. Introduction A Cartan subalgebra, H, of a Lie algebra, G, is a subalgebra, H G, such that a. H is nilpotent, i.e., there is some n such that (H)
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationChapter 1  Matrices & Determinants
Chapter 1  Matrices & Determinants Arthur Cayley (August 16, 1821  January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationIterative Methods for Computing Eigenvalues and Eigenvectors
The Waterloo Mathematics Review 9 Iterative Methods for Computing Eigenvalues and Eigenvectors Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca Abstract: We examine some numerical iterative
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationChapter 7. Lyapunov Exponents. 7.1 Maps
Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationLecture 2. Observables
Lecture 2 Observables 13 14 LECTURE 2. OBSERVABLES 2.1 Observing observables We have seen at the end of the previous lecture that each dynamical variable is associated to a linear operator Ô, and its expectation
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationThe Hadamard Product
The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationThe Basics of FEA Procedure
CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring
More informationDetection of quantum entanglement in physical systems
Detection of quantum entanglement in physical systems Carolina Moura Alves Merton College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2005 Abstract Quantum entanglement
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 3 Linear Least Squares Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign Copyright c 2002. Reproduction
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationORDERS OF ELEMENTS IN A GROUP
ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationMehtap Ergüven Abstract of Ph.D. Dissertation for the degree of PhD of Engineering in Informatics
INTERNATIONAL BLACK SEA UNIVERSITY COMPUTER TECHNOLOGIES AND ENGINEERING FACULTY ELABORATION OF AN ALGORITHM OF DETECTING TESTS DIMENSIONALITY Mehtap Ergüven Abstract of Ph.D. Dissertation for the degree
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationAdditional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman
Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman Email address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents
More informationLinear Least Squares
Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationTill now, almost all attention has been focussed on discussing the state of a quantum system.
Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationLinear Algebra In Dirac Notation
Chapter 3 Linear Algebra In Dirac Notation 3.1 Hilbert Space and Inner Product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex
More informationThe Mathematics of Origami
The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................
More informationNumerical Linear Algebra Chap. 4: Perturbation and Regularisation
Numerical Linear Algebra Chap. 4: Perturbation and Regularisation Heinrich Voss voss@tuharburg.de Hamburg University of Technology Institute of Numerical Simulation TUHH Heinrich Voss Numerical Linear
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationCryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More information1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams
In Neil Salkind (Ed.), Encyclopedia of Research Design. Thousand Oaks, CA: Sage. 00 Matrix Algebra Hervé Abdi Lynne J. Williams Introduction Sylvester developed the modern concept of matrices in the 9th
More information7 Communication Classes
this version: 26 February 2009 7 Communication Classes Perhaps surprisingly, we can learn much about the longrun behavior of a Markov chain merely from the zero pattern of its transition matrix. In the
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationMachine Learning and Pattern Recognition Logistic Regression
Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,
More informationIntroduction to time series analysis
Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationA Introduction to Matrix Algebra and Principal Components Analysis
A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra
More informationQuantum Mechanics I: Basic Principles
Quantum Mechanics I: Basic Principles Michael A. Nielsen University of Queensland I ain t no physicist but I know what matters  Popeye the Sailor Goal of this and the next lecture: to introduce all the
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More information