The Kronecker Product A Product of the Times

Size: px
Start display at page:

Download "The Kronecker Product A Product of the Times"

Transcription

1 The Kronecker Product A Product of the Times Charles Van Loan Department of Computer Science Cornell University

2 The Kronecker Product B C is a block matrix whose ij-th block is b ij C. E.g., [ b11 b 12 b 21 b 22 ] C = b 11C b 12 C b 21 C b 22 C Also called the Direct Product or the Tensor Product

3 Every b ij c kl Shows Up [ b11 b 12 b 21 b 22 ] c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 = b 11 c 11 b 11 c 12 b 11 c 13 b 12 c 11 b 12 c 12 b 12 c 13 b 11 c 21 b 11 c 22 b 11 c 23 b 12 c 21 b 12 c 22 b 12 c 23 b 11 c 31 b 11 c 32 b 11 c 33 b 12 c 31 b 12 c 32 b 12 c 33 b 21 c 11 b 21 c 12 b 21 c 13 b 22 c 11 b 22 c 12 b 22 c 13 b 21 c 21 b 21 c 22 b 21 c 23 b 22 c 21 b 22 c 22 b 22 c 23 b 21 c 31 b 21 c 32 b 21 c 33 b 22 c 31 b 22 c 32 b 22 c 33

4 Basic Algebraic Properties (B C) T = B T C T (B C) 1 = B 1 C 1 (B C)(D F) = BD CF B (C D) C B = (B C) D = (Perfect Shuffle) T (B C)(Perfect Shuffle)

5 Reshaping KP Computations Suppose B,C IR n n and x IR n2. The operation y = (B C)x is O(n 4 ): y = kron(b,c)*x The operation Y = CXB T is O(n 3 ): y = reshape(c*reshape(x,n,n)*b,n,n)

6 Talk Outline 1. The 1800 s Origins: Z 2. The 1900 s Heightened Profile: 3. The 2000 s Future:

7 The 1800 s Z

8 Products and Deltas δ ij Leopold Kronecker ( )

9 The Kronecker Delta: Not as Interesting! U T δ ij V = δij κ 2 (δ ij ) = 1 δ ij

10 Acknowledgement H.V. Henderson, F. Pukelsheim, and S.R. Searle (1983). On the History of the Kronecker Product, Linear and Multilinear Algebra 14, Shayle Searle, Professor Emeritus, Cornell University (right)

11 Scandal! H.V. Henderson, F. Pukelsheim, and S.R. Searle (1983). On the History of the Kronecker Product, Linear and Multilinear Algebra 14, Abstract History reveals that what is today called the Kronecker product should be called the Zehfuss Product. This fact is somewhat appreciated by the modern (numerical) linear algebra community: R.J. Horn and C.R. Johnson(1991). Topics in Matrix Analysis, Cambridge University Press, NY, p A.N. Langville and W.J. Stewart (2004). The Kronecker product and stochastic automata networks, J. Computational and Applied Mathematics 167,

12 Let s Get to the Bottom of This! History Reveals That What is Today Called The Kronecker Product Should Be Called The Zehfuss Product

13 Who Was Zeyfuss? Born Obscure professor of mathematics at University of Heidelberg for a while. Then went on to other things. Wrote papers on determinants... G. Zeyfuss (1858). Uber eine gewasse Determinante, Zeitscrift fur Mathematik und Physik 3,

14 Main Result a.k.a. The Z Theorem If B IR m m and C IR n n then det (B C) = det(b) n det(c) m Proof: Noting that I n B and I m C are block diagonal, take determinants in B C = (B I n )(I m C) = P(I n B)P T (I m C) where P is a perfect shuffle permutation.

15 Zehfuss(1858) = a 1 A 1 a 1 B 1 b 1 A 1 b 1 B 1 c 1 A 1 c 1 B 1 d 1 A 1 d 1 B 1 a 1 A 2 a 1 B 2 b 1 A 2 b 1 B 2 c 1 A 2 c 1 B 2 d 1 A 2 d 1 B 2 a 2 A 1 a 2 B 1 b 2 A 1 b 2 B 1 c 2 A 1 c 2 B 1 d 2 A 1 d 2 B 1 a 2 A 2 a 2 B 2 b 2 A 2 b 2 B 2 c 2 A 2 c 2 B 2 d 2 A 2 d 2 B 2 a 3 A 1 a 3 B 1 b 3 A 1 b 3 B 1 c 3 A 1 c 3 B 1 d 3 A 1 d 3 B 1 a 3 A 2 a 3 B 2 b 3 A 2 b 3 B 2 c 3 A 2 c 3 B 2 d 3 A 2 d 3 B 2 a 4 A 1 a 4 B 1 b 4 A 1 b 4 B 1 c 4 A 1 c 4 B 1 d 4 A 1 d 4 B 1 a 4 A 2 a 4 B 2 b 4 A 2 b 4 B 2 c 4 A 2 c 4 B 2 d 4 A 2 d 4 B 2

16 Zehfuss(1858) p = a 1 b 1 c 1 d 1 a 2 b 2 c 2 d 2 a 3 b 3 c 3 d 3 a 4 b 4 c 4 d 4 und P = A 1 B 1 A 2 B 2 2 2,2 = p 2 4 P Mm = p M P m

17 Hensel (1891) Student in Berlin Maintains that Kronecker presented the Z-theorem in his lectures. K. Hensel (1891). Uber die Darstellung der Determinante eines Systems, welsches aus zwei a irt ist, ACTA Mathematica 14,

18 The 1900 s

19 Muir (1911) Attributes the Z-theorem to Zehfuss. Calls det(b C) the Zehfuss determinant. T. Muir (1911). The Theory of Determinants in the Historical Order of Development, Vols 1-4, Dover, NY.

20 Rutherford(1933) On the Condition that Two Zehfuss Matrices are Equal B C??? = F G If B (m b n b ), C (m c n c ), F (m f n f ), G (m g n g ), then (B C) ij = (F G) ij means B(floor(i/m c ), floor(j/n c )) C(i mod m c,j mod n c ) = F(floor(i/m g ), floor(j/n g )) G(i mod m g,j mod n g )

21 Van Loan(1934) On the Condition that Two Square Zehfuss Matrices are Equal B C??? = F G If then U T b BV b = diag(σ i (B)) U T c CV c = diag(σ i (C)) U T b FV b = α diag(σ i (F)) U T c GV c = 1 α diag(σ i(g))

22 Heightened Profile Beginning in the 60s Some Reasons: Regular Grids Tensoring Low Dimension Ideas Higher Order Statistics Fast Transforms Preconditioners Numerical Multi-linear Algebra

23 Regular Grids (M +1)-by-(N +1) discretization of the Laplacian on a rectangle... A = I M T N + T M I N T 5 =

24 Tensoring Low Dimension Ideas b a f(x) dx n i=1 w i f(x i ) = w T f(x) b1 b2 b3 a 1 a 2 a 3 f(x,y,z) dxdy dz n x i=1 n y j=1 n z k=1 w (x) i w (y) j w (z) k f(x i,y j,z k ) = (w (x) w (y) w (z) ) T f(x y z)

25 Higher Order Statistics E(xx T ) E(x x) E(x x x) Kronecker powers: k A = A A A (k times)

26 Fast Transforms FFT B 4 = F 16 P 16 = B 16 (I 2 B 8 )(I 4 B 4 )(I 8 B 2 ) ω ω 4 ω n = exp( 2πi/n) Haar Wavelet Transform W 2m = [ W m ( 1 1 ) I m ( 1 1 ) ] if m > 1 [ 1 ] if m = 1.

27 Preconditioners If A B C, then B C has potential as a preconditioner. It captures the essence of A. It is easy to solve (B C)z = r. Best Example: A band block Toeplitz with banded Toeplitz blocks. B and C chosen to be band Toeplitz. Nagy, Kamm, Kilmer, Perrone, others

28 Tensor Decompositions/Approximation Given A = A(1:n, 1:n, 1:n, 1:n), find orthogonal Q = [ q 1 q n ] U = [ u 1 u n ] V = [ v 1 v n ] W = [ w 1 w n ] and a core tensor σ so n vec(a) σ ijk,l w i v j u k q l i,j,k,l=1

29 Offspring 1. The Left Kronecker Product 2. The Hadamard Product 3. The Tracy-Singh Product 4. The Khatri-Rao Product 5. The Generalized Kronecker Product 6. The Symmetric Kronecker Product 7. The Bi-Alternate Product

30 Left Kronecker Product Definition: ] [ c11 B c 12 B ] B Left [ c11 c 12 c 21 c 22 = c 21 B c 22 B = C B Fact: If B IR m b n b and C IR m c n c then B Left C = Π T m c, m b m c (B C)Π nc, n b n c Perfect Shuffles

31 The Hadamard Product Definition: b 11 b 12 b 21 b 22 Had c 11 c 12 c 21 c 22 = b 11 c 11 b 12 c 12 b 21 c 21 b 22 c 22 b 31 b 32 c 31 c 32 b 31 c 31 b 32 c 32 B Had C = B. C

32 The Hadamard Product Fact: If à = B C and B,C IR m n, then B Had C = Ã(1:(m+1):m2, 1:(n+1):n 2 ) E.g., b 11 b 12 b 21 b 22 b 31 b 32 c 11 c 12 c 21 c 22 c 31 c 32 = b 11 c 11 b 11 c 12 b 12 c 11 b 12 c 12 b 11 c 21 b 11 c 22 b 12 c 21 b 12 c 22 b 11 c 31 b 11 c 32 b 12 c 31 b 12 c 32 b 21 c 11 b 21 c 12 b 22 c 11 b 22 c 12 b 21 c 21 b 21 c 22 b 22 c 21 b 22 c 22 b 21 c 31 b 21 c 32 b 22 c 31 b 22 c 32 b 31 c 11 b 31 c 12 b 32 c 11 b 32 c 12 b 31 c 21 b 31 c 22 b 32 c 21 b 32 c 22 b 31 c 31 b 31 c 32 b 32 c 31 b 32 c 32

33 The Tracy-Singh Product Definition: B = B TS C = B 11 B 12 B 21 B 22 B 31 B 32 C = [ C11 C 12 C 21 C 22 B 11 C 11 B 11 C 12 B 12 C 11 B 12 C 12 B 11 C 21 B 11 C 22 B 12 C 21 B 12 C 22 B 21 C 11 B 21 C 12 B 22 C 11 B 22 C 12 B 21 C 21 B 21 C 22 B 22 C 21 B 22 C 22 B 31 C 11 B 31 C 12 B 32 C 11 B 32 C 12 B 31 C 21 B 31 C 22 B 32 C 21 B 32 C 22 ]

34 The Khatri-Rao Product Definition: B = B 11 B 12 B 21 B 22 C = C 11 C 12 C 21 C 22 B 31 B 32 B K-R C = C 31 C 32 B 11 C 11 B 12 C 12 B 21 C 21 B 22 C 22 B 31 C 31 B 32 C 32

35 The Khatri-Rao Product Fact: B TS C = B KR C is a submatrix of B TS C B 11 C 11 B 11 C 12 B 12 C 11 B 12 C 12 B 11 C 21 B 11 C 22 B 12 C 21 B 12 C 22 B 11 C 31 B 11 C 32 B 12 C 31 B 12 C 32 B 21 C 11 B 21 C 12 B 22 C 11 B 22 C 12 B 21 C 21 B 21 C 22 B 22 C 21 B 22 C 22 B 21 C 31 B 21 C 32 B 22 C 31 B 22 C 32 B 31 C 11 B 31 C 12 B 32 C 11 B 32 C 12 B 31 C 21 B 31 C 22 B 32 C 21 B 32 C 22 B 31 C 31 B 31 C 32 B 32 C 31 B 32 C 32

36 The Generalized Kronecker Product Regalia and Mitra (1989), Kronecker Products, Unitary Matrices, and Signal Processing Applications, SIAM Review, 31, B 1 B 2 B 3 B 4 gen C = B 1 Left C(1, :) B 2 Left C(2, :) B 3 Left C(3, :) B 4 Left C(4, :)]

37 The Generalized Kronecker Product B 1 B 2 B 3 B 4 GEN { C1 C 2 } = { B1 B 2 { B3 B 4 } gen C 1 } gen C 2

38 The Symmetric Kronecker Product Kronecker Product turns matrix equations into vector equations: CXB T = G (B C) vec(x) = vec(g) The symmetric Kronecker product does the same thing for matrix equations with symmetric solutions: 1 2 ( CXB T + BXC T) = G (symmetric) (B sym C) svec(x) = svec(g) where svec x 11 x 12 x 13 x 12 x 22 x 23 x 13 x 23 x 33 = [ ] T x 11 2x12 x 22 2x13 2x23 x 33

39 Symmetric Kronecker Product Fact: If P = α α α α α α α α = 1/ 2 then vec(x) = P svec(x) and B sym C = P T (B C)P

40 Bi-Alternate Product B C = 1 2 (B C + C B) W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM.

41 The 2000 s Three Predictions

42 Big N Will Mean Big d Will Mean KP A = [ a11 a 12 a 21 a 22 ] [ b11 b 12 b 21 b 22 ] [ z11 z 12 z 21 z 22 ] N = 2 d

43 Think Scalar Think Block Think Tensor If for all 1 m i n we have, n i 1,i 2,i 3,i 4 =1 B(m 1,m 2,m 3,m 4 ) = W(i 1,m 1 )Y (i 2,m 2 )X(i 3,m 3 )Z(i 4,m 4 )A(i 1,i 2,i 3,i 4 ) then B = (W Y ) T A(X Z)

44 Data-Sparse Approximate Factorizations A (B 1 C 1 )(B 2 C 2 )(B 3 C 3 )

45 Det(Log(A)) A (B C)(D E)(F G) log(det(a)) n c log(det(b)) + n b log(det(c)) + n e log(det(d)) + n d log(det(e)) + n g log(det(f)) + n f log(det(g)) See Zehfuss (2058).

1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems 1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Inner product. Definition of inner product

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

More information

1 Introduction to Matrices

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

More information

Continuity of the Perron Root

Continuity of the Perron Root Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

More information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i. Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

More information

R mp nq. (13.1) a m1 B a mn B

R mp nq. (13.1) a m1 B a mn B This electronic version is for personal use and may not be duplicated or distributed Chapter 13 Kronecker Products 131 Definition and Examples Definition 131 Let A R m n, B R p q Then the Kronecker product

More information

Unit 18 Determinants

Unit 18 Determinants Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of

More information

Notes on Matrix Calculus

Notes on Matrix Calculus Notes on Matrix Calculus Paul L. Fackler North Carolina State University September 27, 2005 Matrix calculus is concerned with rules for operating on functions of matrices. For example, suppose that an

More information

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

More information

The Determinant: a Means to Calculate Volume

The Determinant: a Means to Calculate Volume The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

More information

Similarity and Diagonalization. Similar Matrices

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

More information

A note on companion matrices

A note on companion matrices Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod

More information

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

More information

The Hadamard Product

The 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 information

Math 312 Homework 1 Solutions

Math 312 Homework 1 Solutions Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

More information

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

4. MATRICES Matrices

4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

More information

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

More information

Similar matrices and Jordan form

Similar 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 information

Lecture 4: Partitioned Matrices and Determinants

Lecture 4: Partitioned Matrices and Determinants Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying

More information

Decision-making with the AHP: Why is the principal eigenvector necessary

Decision-making with the AHP: Why is the principal eigenvector necessary European Journal of Operational Research 145 (2003) 85 91 Decision Aiding Decision-making with the AHP: Why is the principal eigenvector necessary Thomas L. Saaty * University of Pittsburgh, Pittsburgh,

More information

5. Orthogonal matrices

5. Orthogonal matrices L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal

More information

More than you wanted to know about quadratic forms

More than you wanted to know about quadratic forms CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences More than you wanted to know about quadratic forms KC Border Contents 1 Quadratic forms 1 1.1 Quadratic forms on the unit

More information

Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

More information

Orthogonal Diagonalization of Symmetric Matrices

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

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

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

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

More information

Section 6.1 - Inner Products and Norms

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,

More information

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression

The 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 orthogonal-diagonal-orthogonal type matrix decompositions Every

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

More information

1 Symmetries of regular polyhedra

1 Symmetries of regular polyhedra 1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

More information

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution

SF2940: Probability theory Lecture 8: Multivariate Normal Distribution SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2014 Timo Koski () Mathematisk statistik 24.09.2014 1 / 75 Learning outcomes Random vectors, mean vector, covariance

More information

Factorization Theorems

Factorization 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 information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

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

More information

[1] Diagonal factorization

[1] Diagonal factorization 8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

More information

8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

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:

More information

Brief Review of Tensors

Brief Review of Tensors Appendix A Brief Review of Tensors A1 Introductory Remarks In the study of particle mechanics and the mechanics of solid rigid bodies vector notation provides a convenient means for describing many physical

More information

3. INNER PRODUCT SPACES

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.

More information

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

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

More information

Explicit inverses of some tridiagonal matrices

Explicit inverses of some tridiagonal matrices Linear Algebra and its Applications 325 (2001) 7 21 wwwelseviercom/locate/laa Explicit inverses of some tridiagonal matrices CM da Fonseca, J Petronilho Depto de Matematica, Faculdade de Ciencias e Technologia,

More information

Chapter 6. Orthogonality

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

More information

DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH

DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

More information

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

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

More information

2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant 2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

More information

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

More information

Theory of Matrices. Chapter 5

Theory of Matrices. Chapter 5 Chapter 5 Theory of Matrices As before, F is a field We use F[x] to represent the set of all polynomials of x with coefficients in F We use M m,n (F) and M m,n (F[x]) to denoted the set of m by n matrices

More information

4.6 Null Space, Column Space, Row Space

4.6 Null Space, Column Space, Row Space NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Linear Algebra: Determinants, Inverses, Rank

Linear Algebra: Determinants, Inverses, Rank D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of

More information

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors 1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

More information

The cover SU(2) SO(3) and related topics

The cover SU(2) SO(3) and related topics The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of

More information

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1 (d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

More information

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A. Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

More information

1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams

1 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 information

26. Determinants I. 1. Prehistory

26. Determinants I. 1. Prehistory 26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent

More information

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. 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

More information

Modélisation et résolutions numérique et symbolique

Modélisation et résolutions numérique et symbolique Modélisation et résolutions numérique et symbolique via les logiciels Maple et Matlab Jeremy Berthomieu Mohab Safey El Din Stef Graillat Mohab.Safey@lip6.fr Outline Previous course: partial review of what

More information

Lecture 2 Matrix Operations

Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

More information

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

Lecture Notes on The Mechanics of Elastic Solids

Lecture Notes on The Mechanics of Elastic Solids Lecture Notes on The Mechanics of Elastic Solids Volume 1: A Brief Review of Some Mathematical Preliminaries Version 1. Rohan Abeyaratne Quentin Berg Professor of Mechanics Department of Mechanical Engineering

More information

AXIOMS FOR INVARIANT FACTORS*

AXIOMS FOR INVARIANT FACTORS* PORTUGALIAE MATHEMATICA Vol 54 Fasc 3 1997 AXIOMS FOR INVARIANT FACTORS* João Filipe Queiró Abstract: We show that the invariant factors of matrices over certain types of rings are characterized by a short

More information

9 MATRICES AND TRANSFORMATIONS

9 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 information

equations Karl Lundengård December 3, 2012 MAA704: Matrix functions and matrix equations Matrix functions Matrix equations Matrix equations, cont d

equations Karl Lundengård December 3, 2012 MAA704: Matrix functions and matrix equations Matrix functions Matrix equations Matrix equations, cont d and and Karl Lundengård December 3, 2012 Solving General, Contents of todays lecture and (Kroenecker product) Solving General, Some useful from calculus and : f (x) = x n, x C, n Z + : f (x) = n x, x R,

More information

A Primer on Index Notation

A Primer on Index Notation A Primer on John Crimaldi August 28, 2006 1. Index versus Index notation (a.k.a. Cartesian notation) is a powerful tool for manipulating multidimensional equations. However, there are times when the more

More information

6. Cholesky factorization

6. Cholesky factorization 6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

More information

UNCOUPLING THE PERRON EIGENVECTOR PROBLEM

UNCOUPLING THE PERRON EIGENVECTOR PROBLEM UNCOUPLING THE PERRON EIGENVECTOR PROBLEM Carl D Meyer INTRODUCTION Foranonnegative irreducible matrix m m with spectral radius ρ,afundamental problem concerns the determination of the unique normalized

More information

DATA ANALYSIS II. Matrix Algorithms

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

More information

Continued Fractions and the Euclidean Algorithm

Continued 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 information

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

More information

Linear Algebra Notes

Linear Algebra Notes Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note

More information

RINGS WITH A POLYNOMIAL IDENTITY

RINGS WITH A POLYNOMIAL IDENTITY RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in

More information

The Inverse of a Square Matrix

The Inverse of a Square Matrix These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

Inner products on R n, and more

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 + +

More information

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

More information

Solution to Homework 2

Solution to Homework 2 Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

More information

Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph

Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph FPSAC 2009 DMTCS proc (subm), by the authors, 1 10 Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph Christopher R H Hanusa 1 and Thomas Zaslavsky 2 1 Department

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

Tensors on a vector space

Tensors on a vector space APPENDIX B Tensors on a vector space In this Appendix, we gather mathematical definitions and results pertaining to tensors. The purpose is mostly to introduce the modern, geometrical view on tensors,

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

Matrix algebra for beginners, Part I matrices, determinants, inverses

Matrix algebra for beginners, Part I matrices, determinants, inverses Matrix algebra for beginners, Part I matrices, determinants, inverses Jeremy Gunawardena Department of Systems Biology Harvard Medical School 200 Longwood Avenue, Cambridge, MA 02115, USA jeremy@hmsharvardedu

More information

Finite dimensional C -algebras

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

More information

BILINEAR FORMS KEITH CONRAD

BILINEAR FORMS KEITH CONRAD BILINEAR FORMS KEITH CONRAD The geometry of R n is controlled algebraically by the dot product. We will abstract the dot product on R n to a bilinear form on a vector space and study algebraic and geometric

More information

Stiffness Matrices of Isoparametric Four-node Finite Elements by Exact Analytical Integration

Stiffness Matrices of Isoparametric Four-node Finite Elements by Exact Analytical Integration Stiffness Matrices of Isoparametric Four-node Finite Elements by Exact Analytical Integration Key words: Gautam Dasgupta, Member ASCE Columbia University, New York, NY C ++ code, convex quadrilateral element,

More information

A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

More information

Introduction to Matrices for Engineers

Introduction to Matrices for Engineers Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

More information

Generalized Inverse Computation Based on an Orthogonal Decomposition Methodology.

Generalized Inverse Computation Based on an Orthogonal Decomposition Methodology. International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006 Generalized Inverse Computation Based on an Orthogonal Decomposition Methodology. Patricia

More information

MATH 551 - APPLIED MATRIX THEORY

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

More information

Quantum Computers. And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015. Quantum Computers

Quantum Computers. And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015. Quantum Computers Quantum Computers And How Does Nature Compute? Kenneth W. Regan 1 University at Buffalo (SUNY) 21 May, 2015 1 Includes joint work with Amlan Chakrabarti, U. Calcutta If you were designing Nature, how would

More information

On the D-Stability of Linear and Nonlinear Positive Switched Systems

On the D-Stability of Linear and Nonlinear Positive Switched Systems On the D-Stability of Linear and Nonlinear Positive Switched Systems V. S. Bokharaie, O. Mason and F. Wirth Abstract We present a number of results on D-stability of positive switched systems. Different

More information