# The Kronecker Product A Product of the Times

Save this PDF as:

Size: px
Start display at page:

## 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 Ã = 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).

### 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

### 1 Determinants. Definition 1

Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### 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

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### 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

### 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

### Characterizations are given for the positive and completely positive maps on n n

Special Classes of Positive and Completely Positive Maps Chi-Kwong Li and Hugo J. Woerdemany Department of Mathematics The College of William and Mary Williamsburg, Virginia 387 E-mail: ckli@cs.wm.edu

### 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

### 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

### Linear Algebra of Magic Squares

Linear Algebra of Magic Squares Michael Lee Central Michigan University Mt Pleasant, MI 48859 email: lee4m@cmichedu Elizabeth Love Howard University Washington, DC 0001 email: lle 17@hotmailcom Elizabeth

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### 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

### 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

### 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

### Mathematics of Cryptography Part I

CHAPTER 2 Mathematics of Cryptography Part I (Solution to Odd-Numbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.

### 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)(

### 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

### 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

### 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

### 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

### 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

### EC9A0: Pre-sessional Advanced Mathematics Course

University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

### 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

### 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

### 9 Matrices, determinants, inverse matrix, Cramer s Rule

AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

### 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

### 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

### 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

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

### D(N) := 1 N 1 [NJ N I N ],

09.09.2 Inverse elementary divisor problems for nonnegative matrices Abstract. The aim of this paper is to answer three questions formulated by Minc in his two papers and book on the problem of prescribed

### 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:

### 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,

### 2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

### 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 +

### 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

### 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

### We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

### 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

### Sign pattern matrices that admit M, N, P or inverse M matrices

Sign pattern matrices that admit M, N, P or inverse M matrices C Mendes Araújo, Juan R Torregrosa CMAT - Centro de Matemática / Dpto de Matemática Aplicada Universidade do Minho / Universidad Politécnica

### 1 Vector Spaces and Matrix Notation

1 Vector Spaces and Matrix Notation De nition 1 A matrix: is rectangular array of numbers with n rows and m columns. 1 1 1 a11 a Example 1 a. b. c. 1 0 0 a 1 a The rst is square with n = and m = ; the

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### 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

### 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

### 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,

### 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

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

### Matrices, transposes, and inverses

Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4

### 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

### 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

### 5.3 Determinants and Cramer s Rule

290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

### Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

### 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,

### 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.

### Mathematics Notes for Class 12 chapter 3. Matrices

1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

### LINEAR ALGEBRA. September 23, 2010

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

### 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,

### Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

### [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:

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67

Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of

Math 210 Quadratic Polynomials Jerry L. Kazdan Polynomials in One Variable. After studying linear functions y = ax + b, the next step is to study quadratic polynomials, y = ax 2 + bx + c, whose graphs

### Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

### Mathematics 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

### Determinants. Chapter Properties of the Determinant

Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation

### 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

### 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

### 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

### 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

### 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.

### 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,

### 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:

### 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

### Divisibility of the Determinant of a Class of Matrices with Integer Entries

Divisibility of the Determinant of a Class of Matrices with Integer Entries Sujan Pant and Dennis I. Merino August 8, 010 Abstract Let x i = n j=1 a ij10 n j, with i = 1,..., n, and with each a ij Z be

### 9. Numerical linear algebra background

Convex Optimization Boyd & Vandenberghe 9. Numerical linear algebra background matrix structure and algorithm complexity solving linear equations with factored matrices LU, Cholesky, LDL T factorization

### 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

### 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,

### 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

### 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.

### 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)

### 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

### 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

### 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,

### 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

### Math 2331 Linear Algebra

2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University

### 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

### 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