# Lecture 2: Essential quantum mechanics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 1/46 Quantum information and computing Lecture 2: Essential quantum mechanics Jani-Petri Martikainen jamartik Department of Physical Sciences University of Helsinki

2 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 2/46 Disclaimer This lecture is going to extremely boring, for everyone involved. The faster we go through this, the faster we can go on with our lives. I promise to wake you up in the end.

3 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 3/46 Vector spaces Basic objects are vector spaces, we are interested in C n, the space of all n-tuples of complex numbers (z 1...z n ) elements of vector space are called vectors, we might use notation z 1. z n (1) Addition takes vectors to another vectors z 1 z 1 z 1 + z (2) z n z n z n + z n

4 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 4/46 Multiplication by a scalar Vector spaces z z 1. zz 1. (3) z n zz n In QM the vector is usually denoted with a ket-notation ψ, where ψ is used to label the vector in question. zero vector: ψ + 0 = ψ, note we do not use 0 notation for the zero vector, since in physics this usually means something different. To make typing easier: (z 1...z n ) implies the column matrix with entries z 1...z n W is a vector subspace of V if it is a vector space and closed under scalar multiplication and addition.

5 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 5/46 Bases and linear independence a vector space is spanned by a set of vectors v 1,..., v n such that any vector in V can be written as a v = i a i v i For example C 2 : v 1 = (1, 0) and v 2 = (0, 1) basis vectors are not unique: v 1 = (1, 1)/ 2 and v 2 = (1, 1)/ 2 would also be OK vectors v 1,..., v n are linearly dependent if a 1 v 1 + a n v n = 0 (4) for some a i, with a i 0 for at least one i Otherwise, vectors are linearly independent and can form a basis of the vector space

6 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 6/46 Bases...operators and matrices Number of basis elements is the dimension of the vector space we will just deal with finite dimensional vector spaces a linear operator between vector spaces V and W is defined as a function A : V W which is linear in its inputs i.e. A( i a i v i ) = i Identity operator I: I v v for all v in V zero operator: 0 v = 0 a i A v i (5) Once the action of linear A on the basis vectors is known, we know the action for all vectors in V

7 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 7/46 Operators and matrices V,W,X are vector spaces and A : V W B : W X are linear operators. BA is the composition of B with A, defined by (BA)( v ) = B(A v ) matrix representations of linear operators: m by n complex matrix with entries A ij sends a vector in C n into C m under matrix multiplication How to find a matrix representation for a linear operator?: Linear operator (from V to W ) defined through its action on the basis vectors A v j = i A ij w i (6)

8 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 8/46 Operators and matrices The matrix entries A ij form the matrix representation of the linear operator A...note: had to define input and output vector space basis. Pauli matrices: σ 0 = I = [ ] σ x = X = [ ] (7) σ y = Y = [ 0 i i 0 ] σ z = Z = [ ] (8)

9 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 9/46 Inner product inner product takes two input vectors and produces a complex number: ( v, w ) In quantum mechanics inner product of v and w is denoted by v w The bra-vector v is the dual vector to v the matrix representation of dual vectors is just a row vector

10 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 10/46 Inner product requirements 1. (, ) is linear in the second argument: ( v, i λ i w i ) = i λ i ( v, w ) (9) 2. ( v, w ) = ( w, v ) 3. ( v, v ) 0 with equality if only if v = 0 4. For example, C n : ((y 1 y n ), (z 1 z n )) = i y i z i = [y 1 y n] z 1. z n (10)

11 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 11/46 Inner product Hilbert space: vector space with inner product Vectors are orthogonal if their inner product is zero. norm: v v v vector is normalized if its norm is 1 orthonormal basis is a set of normalized orthogonal basis vectors From now on matrix representations of linear operators are assumed to be defined with respect to orthonormal bases.

12 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 12/46 Gram-Schmidt Gram-Schmidt procedure to construct an orthonormal basis: w 1,... w n is some basis set. Define v 1 w 1 / w 1 for 1 k n 1 define the new basis vector inductively v k+1 w k+1 k i=1 v i w k+1 v i w k+1 k i=1 v i w k+1 v i (11)

13 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 13/46 Inner product...outer product if w = w i i and v = v j j representation of vectors with respect to orthonormal basis then since i j = δ ij v w = i v i w i Outer product operator w v takes a vector from V to W : ( w v )( v ) = v v w (12) take v = i v i i, then ( i i ) v = i i v = v (13) so i i = I which is the completeness relation

14 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 14/46...Cauchy-Schwartz inequality Can use completeness to represent operators in the outer product notation. Outer product representation of A: A = I W AI V = ij w j w j A v i v i = ij w j A v i w j v i Cauchy-Schwartz inequality: v w 2 v v w w (14) Proof (idea): using Gram-Schmidt decomposition construct an orthonormal basis i so that w / w w is the first element. Use completeness i i = I and drop some non-negative terms from v v w w = v i i v w w... (15)

15 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 15/46 Eigenvectors and eigenvalues Diagonal representation or spectral decomposition of A: A = λ i i i (16) where λ i are the eigenvalues and i are the eigenvectors Operator is diagonalizable if it has a diagonal representation [ ] 1 0 For example: Z = = If several eigenvalues are the same...eigenspace is degenerate

16 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 16/46 Hermitian operators Hermitian conjugate or adjoint A of the operator A: ( v,a w ) = (A v, w ) (AB) = B A define v v...(a v ) = v A In matrix representation: A = (A ) T where T indicates transpose An operator which is its own adjoint is Hermitian or self-adjoint

17 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 17/46 Projectors Projectors are an important class of Hermitian operators Suppose W is the k-dimensional subspace of n-dimensional vector space V. We can construct a basis 1,..., n for V so that 1,..., k is the basis for W Projector into subspace W : P = k i=1 i i Orthogonal complement: Q = I P

18 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 18/46 Unitary operators and tensor products Unitary operator if U U = I = UU Important (among other things) because they preserve inner products between vectors (U v,u w ) = v U U w = v I w = v w (17) tensor product: put vector spaces together to great bigger vector spaces V W... the elements of V W are linear combinations of tensor products v w of elements v in V and w in W. If i and j are orthonormal bases then i j is a basis for V W...we will often abbreviate i j = ij

19 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 19/46 Tensor products properties 1. z( v w ) = (z v ) w = v (z w ) 2. ( v 1 + v 2 ) w = v 1 w + v 2 w 3. v ( w 1 + w 2 ) = v w 1 + v w 2 A and B operators in V and W, define A B( v w ) A v B w (18) ensures linearity

20 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 20/46 Tensor products A is m n matrix and B is p q matrix A 11 B A 12 B A 1n B A A B 21 B A 22 B A 2n B... A m1 B A m2 B A mn B. (19) this is mp nq beast

21 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 21/46 Tensor product example [ 1 2 ] [ 2 3 ] = (20) X Y = [ 0 Y 1 Y 1 Y 0 Y ] = i 0 0 i 0 0 i 0 0 i (21)

22 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 22/46 Operator functions If A = a a a then f(a) = f(a) a a (22) For example, exp(θz) = [ e θ 0 0 e θ ] (23) Trace: Tr(A) = i A ii Trace is invariant under similarity transformation i.e. when A UAU with U unitary Simple example on blackboard...

23 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 23/46 Commutators and anti-commutators Commutator: [A,B] = AB BA anti-commutator: {A,B} = AB + BA If hermitian A and B commute i.e. [A,B] = 0, then there exists an orthonormal basis in which both A and B are diagonal. for Pauli matrices: [X,Y ] = 2iZ, [Y,Z] = 2iX, and [Z,X] = 2iY

24 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 24/46 Quantum mechanics On its own QM does not tell you what laws of a physical system must obey. It provides a mathematical and conceptual framework for the development of such laws. Postulates of QM were derived after a long process of trial and error...lots of guessing involved so don t be surprised if the motivation is not always clear.

25 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 25/46 Quantum mechanics Postulate 1: A system is completely described by its state vector in the systems Hilbert space (i.e. unit vector in the state space) qubit has a 2-dimensional state space with basis 0 and 1. The state vector is generally ψ = a 0 + b 1 What the systems Hilbert space is, is not always clear. That depends...i.e. where do you draw the line where the system ends? Weird classically: we cannot directly observe the state vector.

26 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 26/46 Quantum mechanics Postulate 2: Evolution of a closed system is described by a unitary transformation. ψ(t 2 ) = U(t 2,t 1 ) ψ(t 1 ) (24) QM does not tell what U is...that depends... some examples X is called a bit flip: 0 1 and 1 0, on the other hand Z is called a phase flip: 1 1 Hadamard gate: [ H = ] (25)

27 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 27/46 Quantum mechanics Postulate 2 : Evolution of a closed system is described by the Schrödinger equation i ψ t = H ψ (26) spectral decomposition H = E E E is the system Hamiltonian E are the energy eigenvalues Note: even non-closed systems can be sometimes described (or well approximated) by a Hamiltonian. For example, under some conditions an atom in a laser field can be described by a Hamiltonian which can be tuned experimentally.

28 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 28/46 Quantum mechanics: Measurement Postulate 3: Measurements described by a measurement operators {M m }, where index m refers to the measurement outcome that may occur. If ψ is the state just before the measurement then the probability of an outcome m is and after the measurement p(m) = ψ M mm m ψ (27) ψ M m ψ ψ M mm m ψ (28) Completeness of measurement operators: m M mm m = I

29 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 29/46 Quantum mechanics: Measurement Completeness simply implies that the probabilities of the measurement outcomes must sum to 1 Measure a qubit in its computational basis: M 0 = 0 0 and M 1 = 1 1 p(0) = ψ M 0 M 0 ψ = a 2, M 0 ψ / a = a/ a 0 Perhaps it is possible to derive the postulate 3 from the 1st and the 2nd postulate. Seems likely, but to prove it is hard. Classical things are easy to distinguish. In QM distinguishing reliably non-orthogonal state vectors is fundamentally impossible ψ 2 = α ψ 1 + β φ

30 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 30/46 Projective measurements Special case of the general measurement Projective measurement is described by an observable, M, a Hermitian operator. This observable can be decomposed as M = m mp m, (29) where P m is a projector onto eigenspace of M with eigenvalue m Measurement outcomes correspond to the eigenvalues m, probability of m is p(m) = ψ P m ψ and just after measurement the state is projected into P m ψ p(m) (30)

31 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 31/46 Projective measurements Easy to calculate average values of measurements: E(M) = mp(m) = m ψ P m ψ = ψ (mp m ) ψ = ψ M ψ (31)

32 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 32/46 Heisenberg s uncertainty principle A and B are Hermitian and suppose ψ AB ψ = x + iy. Then ψ [A,B] ψ = 2iy and ψ {A,B} ψ = 2x This implies: ψ [A,B] ψ 2 + ψ {A,B} ψ 2 = 4 ψ AB ψ 2 (32) By the Cauchy-Schwarz inequality: ψ AB ψ 2 ψ A 2 ψ ψ B 2 ψ...combine with Eq. 32 and drop a non-negative term ψ [A,B] ψ 2 4 ψ A 2 ψ ψ B 2 ψ (33)

33 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 33/46 Heisenberg s uncertainty principle More normal form: suppose C and D are observables and substitute A = C C and B = D D we obtain the Heisenberg s uncertainty principle (C) (D) [C,D], (34) 2 where (C) = (C C ) 2 is the standard deviation Example, operators X and Y measured for the quantum state 0 : [X,Y ] = 2iZ so (X) (Y ) 0 Z 0 = 1 (35)

34 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 34/46 POVM measurement Measurement postulate has two elements: a) gives a rule to describe measurement statistics b) gives a rule for the post-measurement state For some applications, post-measurement state is not of great interest. Probabilities for the measurement outcomes crucial. For example, experiment where the system is measured once at the end of the experiment Analysis in terms of the POVM (positive-operator-valued-measure) formalism Define E m M mm m whose expectation value gives the probability for the outcome m

35 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 35/46 POVM measurement E m is a positive operator (probabilities are positive) and Em = I E m are known as the POVM elements associated with the experiment and the complete set {E m } is known as the POVM For a projective measurement: P m P m = δ mm P m and the POVM elements are the same as the measurement operators, E m = P mp m = P m Note: Projective measurement is repeatable i.e. if you measure once you project into a state which does not change when you repeat the measurement. I.e. you will always measure the same result.

36 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 36/46 POVM measurement The repeatability suggests than many measurements in QM are NOT projective measurements. (use silvered screen to measure the position of the photon...photon destroyed..cannot repeat the measurement of the photons position) Example: Alice gives Bob a qubit which is either ψ 1 = 0 or ψ 2 = ( )/ 2. Bob can never be absolutely certain of which state he received. However, he can perform a measurement which distinguishes the states some of the time, but NEVER makes an error of mis-identification. Consider POVM with elements: E 1 = 2/(1 + 2) 1 1, E 2 = 2/(1 + 2)( 0 1 )( 0 1 )/2, and E 3 = I E 1 E 2

37 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 37/46 POVM measurement These operators are complete and positive...povm OK. Suppose Bob got ψ 1 = 0 and he performs a measurement described by the POVM {E 1,E 2,E 3 }. There is zero probability for outcome E 1 since ψ 1 E 1 ψ 1 = 0...Bob knows that if the outcome is E 1 he must have got ψ 2 from Alice. Also, since ψ 2 is orthogonal with ( 0 1 )/ 2 outcome E 2 must mean he received ψ 1 Sometimes, Bob obtains an outcome E 3 in which case he cannot infer anything about the states identity However! He never makes a mistake in identification.

38 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 38/46 Phase If we have a state exp(iθ) ψ we will find that the measurement statistics of this state and the state ψ are identical. For this reason, from the observational point of view global phase factor plays no role. Relative phase is different: consider states ( )/ 2 and ( 0 1 )/ 2. Magnitude of the amplitudes are the same, but differ in sign (or more generally by a phase factor exp(iθ)) Relative phases may vary from amplitude to amplitude and the concept of a relative phase is basis-dependent. This means that states with different relative phases can give rise to different measurement statistics.

39 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 39/46 QM: Composite systems Postulate 4: The state space of a composite system is the tensor product of the component system state spaces. If we have systems 1...n and systems are prepared in ψ i with i = 1...n, then the joint state of the system is ψ 1 ψ 2 ψ n Heuristically: It seems natural that if A is permissible state in A and B is permissible state in B, then A B should be permissible in AB. This combined with the superposition principle (that is if states x and y are OK, the also α x + β y is OK) gives you the tensor product postulate. We will use a subscript notation so that X 2 for example refers to the Pauli σ x acting on the 2nd qubit.

40 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 40/46 General measurement Lets show that we can implement general measurements with unitary evolution and projective measurements. Suppose we have a quantum state in state space Q and we wish to perform measurement described by measurement operators M n on the system Q Introduce an ancilla system, with state space M with an orthonormal basis states m with one-to-one correspondence with the possible measurement outcomes. Think of ancilla system as a mathematical device or as an extra physical quantum system which has a state space with the required properties.

41 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 41/46 General measurement Let 0 be any fixed state of M and define U through U ψ 0 = m M m ψ m (36) Using orthonormality of m states and the completeness relation m M mm m = I we see that U preserves inner products: φ 0 U U ψ 0 = m,m φ M mm m ψ m m = m φ M mm m ψ = φ ψ (37)

42 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 42/46 General measurement From this it follows that U can be extended to a unitary operator on the space Q M, which we also denote by U. Suppose we perform a projective measurement on the two systems described by P m = I Q m m...outcome m with probability p(m) = ψ 0 U P m U ψ 0 = m,m ψ M m m (I m m )M m ψ m = ψ M mm m ψ (38) as given by the measurement postulate!

43 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 43/46 General measurement The joint state after the measurement of m is given by P m U ψ 0 ψ U P m U ψ = M m ψ m ψ M mm m ψ (39) just as described by the measurement postulate! Thus unitary dynamics, projective measurements, and the ability to introduce ancilla systems, together allow any general measurement of the form described by the measurement postulate.

44 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 44/46 Entangled states Consider a state ψ = (40) There are NO single qubit states a and b such that ψ = a b...the state is then called entangled Entangled states play a crucial role in quantum computation and information EPR thought experiment based on the possibility of entangled states with parts separated by a large distance. Measurement of, for example, 0 at one end, immediately implies that the one must have a state 0 at the other end.

45 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 45/46 Quantum mechanics We cannot directly observe the state vector. It is as if there is a hidden world in QM which we can access only imperfectly. Observing the state vector, typically changes it. (Play tennis, and each time you look at the ball, its position changes.) Bell s inequalities showed that we are stuck with counter-intuitive nature of QM. Which is GREAT!

46 Department of Physical Sciences, University of Helsinki kvanttilaskenta/ p. 46/46 Exercises There are exercises next week! Download from the course web page...return by 1600 on Monday. WAKE UP! WAKE UP! WAKE UP!

### 2. 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

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### Hilbert Space Quantum Mechanics

qitd114 Hilbert Space Quantum Mechanics Robert B. Griffiths Version of 16 January 2014 Contents 1 Introduction 1 1.1 Hilbert space............................................. 1 1.2 Qubit.................................................

### MATH 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

### Quantum 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

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 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,

### Quick 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...........................

### Introduction to Quantum Information Theory. Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT)

Introduction to Quantum Information Theory Carlos Palazuelos Instituto de Ciencias Matemáticas (ICMAT) carlospalazuelos@icmat.es Madrid, Spain March 013 Contents Chapter 1. A comment on these notes 3

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### Linear 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

### 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 4-5) 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

### α = 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

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### 0.1 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

### Vector 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 non-empty

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### BOX. 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,

### Linear 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

### 1 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

### October 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix

Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,

### Recall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.

ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?

### Applied Linear Algebra I Review page 1

Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

### Mixed 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

### Linear Algebra: Vectors

A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector

### Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

### Summary 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

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### Hermitian 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

### BANACH 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

### Notes 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,

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

### 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:

### 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)

Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

### WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

### Linear 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

### Numerical Methods I Eigenvalue Problems

Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)

### Lecture 1: Schur s Unitary Triangularization Theorem

Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### Finite dimensional C -algebras

Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

### 1 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

### Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

### Bra-ket notation - Wikipedia, the free encyclopedia

Page 1 Bra-ket notation FromWikipedia,thefreeencyclopedia Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract

### Orthogonal 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

### A 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

### Finite Dimensional Hilbert Spaces and Linear Inverse Problems

Finite Dimensional Hilbert Spaces and Linear Inverse Problems ECE 174 Lecture Supplement Spring 2009 Ken Kreutz-Delgado Electrical and Computer Engineering Jacobs School of Engineering University of California,

### Physics 221A Spring 2016 Notes 1 The Mathematical Formalism of Quantum Mechanics

Copyright c 2016 by Robert G. Littlejohn Physics 221A Spring 2016 Notes 1 The Mathematical Formalism of Quantum Mechanics 1. Introduction The prerequisites for Physics 221A include a full year of undergraduate

### Quantum Physics II (8.05) Fall 2013 Assignment 4

Quantum Physics II (8.05) Fall 2013 Assignment 4 Massachusetts Institute of Technology Physics Department Due October 4, 2013 September 27, 2013 3:00 pm Problem Set 4 1. Identitites for commutators (Based

### Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round \$200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### For the case of an N-dimensional 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. on-hermitian

### 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

### Notes on Quantum Mechanics. Finn Ravndal Institute of Physics University of Oslo, Norway

Notes on Quantum Mechanics Finn Ravndal Institute of Physics University of Oslo, Norway e-mail: finnr@fys.uio.no v.4: December, 2006 2 Contents 1 Linear vector spaces 7 1.1 Real vector spaces...............................

### Detection 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

### UNIT 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

### MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### 3. 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.

### MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α

### Homework One Solutions. Keith Fratus

Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product

### Contents Quantum states and observables

Contents 2 Quantum states and observables 3 2.1 Quantum states of spin-1/2 particles................ 3 2.1.1 State vectors of spin degrees of freedom.......... 3 2.1.2 Pauli operators........................

### Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

### Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

### Quantum Computing. Robert Sizemore

Quantum Computing Robert Sizemore Outline Introduction: What is quantum computing? What use is quantum computing? Overview of Quantum Systems Dirac notation & wave functions Two level systems Classical

### A Concise Text on Advanced Linear Algebra

A Concise Text on Advanced Linear Algebra This engaging textbook for advanced undergraduate students and beginning graduates covers the core subjects in linear algebra. The author motivates the concepts

### (January 14, 2009) End k (V ) End k (V/W )

(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### 15.062 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

### Introduction 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

### DATA 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

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

### Linear 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

### MATH 304 Linear Algebra Lecture 24: Scalar product.

MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent

### Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8

Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e

### ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor

### Inner product. Definition of inner product

Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### 3 Orthogonal Vectors and Matrices

3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### 7 - Linear Transformations

7 - Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure

### 8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, App-I Li. 7 1 4 Ga. 4 7, 6 1,2

### Lecture Notes for Ph219/CS219: Quantum Information and Computation Chapter 2. John Preskill California Institute of Technology

Lecture Notes for Ph19/CS19: Quantum Information and Computation Chapter John Preskill California Institute of Technology Updated July 015 Contents Foundations I: States and Ensembles 3.1 Axioms of quantum

### The Sixfold Path to understanding bra ket Notation

The Sixfold Path to understanding bra ket Notation Start by seeing the three things that bra ket notation can represent. Then build understanding by looking in detail at each of those three in a progressive

### Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20

Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical

### Inner products on R n, and more

Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +

### Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz

Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional

### Lecture 18 - Clifford Algebras and Spin groups

Lecture 18 - Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning

### Quantum Computation CMU BB, Fall 2015

Quantum Computation CMU 5-859BB, Fall 205 Homework 3 Due: Tuesday, Oct. 6, :59pm, email the pdf to pgarriso@andrew.cmu.edu Solve any 5 out of 6. [Swap test.] The CSWAP (controlled-swap) linear operator

### 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

### Vector algebra Christian Miller CS Fall 2011

Vector algebra Christian Miller CS 354 - Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority

### NOTES 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

### Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved.

Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved. 1 Sum, Product and Transpose of Matrices. If a ij with