Chapter 5. Matrices. 5.1 Inverses, Part 1


 Joella Reynolds
 1 years ago
 Views:
Transcription
1 Chapter 5 Matrices The classification result at the end of the previous chapter says that any finitedimensional vector space looks like a space of column vectors. In the next couple of chapters we re going to take this further by showing that rewriting vectors as column vectors turns linear transformations into matrices. Before we can do this, we need to collect some information and set up some notation about matrices. Some of this stuff you will have already seen, but maybe not in this generality. I am not going to remind you about how and when you can multiply matrices, or all of the basic properties of matrix multiplication, such as associativity: ABC = ABC whenever this makes sense. Instead, we ll concentrate on the notion of an inverse and the determinant for square matrices. 5.1 Inverses, Part 1 Recall that for a positive integer n, the identity matrix I n is the n n square matrix with 1s on the main diagonal and 0s elsewhere. Eg., I 1 = 1, I 2 =, I 3 = 0 1 0, etc The identity matrix gets its name because given any n n matrix A with entries in a field K, we have AI n = I n A = A, so I n acts as an identity element for the set of n n matrices under multiplication. If the context is clear, then we sometimes drop the n and write I for the identity matrix. In analogy with the usual sort of multiplication eg. for numbers in a field, we say that an n n matrix A with entries in K is invertible if there exists a n n matrix B with AB = I = BA. Lemma 5.1. If A is invertible, then the inverse of A is unique. Hence we are safe writing A 1 for the inverse of an invertible matrix A. Proof. Suppose B and C are two inverses of A, so AB = I = BA and AC = I = CA. Then B = IB = CAB = CAB = CI = C. 29
2 Examples 5.2. Here are some basic examples: i The matrix has inverse. 1 4 ii A general 2 2 matrix A = γ δ only if αδ βγ 0 K. If αδ βγ 0, then A 1 = 1 αδ βγ you can verify by direct calculation with entries in a field K is invertible if and δ β, which γ α iii Any n n matrix A with entries in K can be viewed as a linear transformation from the space K n to itself. Then A is invertible if and only if it is an isomorphism when viewed as a linear transformation. 5.2 Determinants The function αδ βγ on the entries of a 2 2 matrix A = given in Example γ δ 5.2ii above is called the determinant. This important function can be defined for any square matrix. In order to do that, we first need to recall a couple of very basic things about symmetric groups. Definition 5.3. Given a positive integer n, the symmetric group S n is the set of bijections from the set {1,2,...,n} to itself. It has the following basic properties among others: i The order of S n is n!. ii The elements of S n can be written as products of disjoint cycles, for example: σ = is the element of S 6 which sends 1 5, 5 4, 4 1, 2 3, 3 2, 6 6. iii There is a map sgn : S n {1, 1} called signature, defined as follows: for σ S n, write σ as a product of disjoint cycles. Let l 1,...,l r be the lengths of the cycles. Then sgnσ = r i=1 1l i 1. With this in hand, we can define determinants. Definition 5.4. Let A = α ij be an n n matrix with entries in K. Then we define the determinant of A, denoted deta, as follows: deta := σ S n sgnσ n α σj,j. The function det maps matrices with entries in K to elements of K. 30
3 Examples 5.5. The definition of the determinant is not easy to get to grips with without doing some examples. Here are a couple: i Let s check that the two definitions coincide for 2 2 matrices. Let A = γ δ as before. Then, in the notation of the definition above, we have α 11 = α, α 12 = β, α 21 = γ and α 22 = δ. Further, we are interested in the symmetric group S 2 = {12,12}. Then the formula in Definition 5.4, together with the definition of the signature sgn, gives deta = sgn12α 11 α 22 +sgn12α 21 α 12 = αδ+ 1 1 γβ = αδ βγ, as we had before. ii Working over the real numbers, set A = 1 2 π e Then we re in the case n = 3, so we have to consider So. S 3 = {123, 123, 132, 231, 123, 132} with signatures deta = α 11 α 22 α 33 α 21 α 12 α 33 α 31 α 22 α 13 α 11 α 32 α 23 +α 21 α 32 α 13 +α 31 α 12 α 23 = π e 1 = 3 8 3π 2 6e 1. We now give an extremely important result about determinants: the determinant function is multiplicative. Lemma 5.6. IfAandB aren nmatricesoverafieldk, thendetab = detadetb. Remark. For such a fundamental result, the proof is rather tricky to get at. You can, in theory, jump straight in and try to manipulate the various determinants involved, but this is very longwinded. There s a shorter proof in the appendix to this chapter, but it is not examinable. For your piece of mind, you might at least want to check the result directly for 2 2 matrices and if you re brave 3 3. In any case, you should certainly remember the result! Corollary 5.7. If A is invertible, then deta 0 and deta 1 = deta 1. 31
4 Proof. If A 1 exists, then we have by Lemma 5.6, which gives the result. 1 = deti = detaa 1 = detadeta 1, This result shows that the determinant is intimately related to the existence of the inverse for a matrix. There are lots and lots of different ways of calculating determinants. One that you may be familiar with is the Laplace expansion or cofactor expansion: Definition 5.8. Let A = α ij be an n n matrix with entries in K. For each 1 i,j n we can obtain an n 1 n 1 matrix by deleting the i th row and j th column of A. The determinant of this matrix is called the i,jminor of A, denoted M ij. The i,j cofactor of A is then defined to be C ij := 1 i+j M ij. Then the Laplace expansion along the i th row is given by the formula: deta = α ij C ij. Similarly, the Laplace expansion along the j th column is: deta = α ij C ij. i=1 Note that the formula in each case looks very similar; however, in the row expansion formula i is fixed and the js are changing, whereas in the column expansion formula j is fixed and the is are changing. It is an exercise to show that the different Laplace expansions all agree, and that they agree with the definition of the determinant. The point of the Laplace expansion is that it reduces the problem of finding an n n determinant to the problem of finding several n 1 n 1 determinants. Therefore, in principle, you can find any determinant by applying this formula recursively until you get down to matrices which are small enough to deal with easily. Example 5.9. As usual, an example will help. Let s do the matrix 1 2 π A = 0 1 e from Example 5.5 again. Looking at Laplace s formula, it is often sensible to choose a row or column which has 0s in it, because that reduces the number of calculations. Let s expand along the second row, so the formula becomes: deta = α 2j C 2j. 32
5 Since α 21 = 0, we don t need to worry about the first term which is why I chose the second row. Now C 22 = π det = π so deta = C 23 = 1 5 det 5.3 Inverses Part = π 6e 1 = 3 8 3π 2 6e 1, which is the same as we had before. We know that an invertible matrix must have nonzero determinant. In this section, we show the converse is true, and write down the general form for the inverse of a matrix. Definition Suppose A is an n n matrix with entries in K. The cofactor matrix of A is the matrix whose i,jentry is the i,jcofactor of A, that is cofa := C ij. The adjugate matrix of A is the transpose of the cofactor matrix, that is adja := cofa has i,jentry the j,icofactor of A. Theorem Let A be an n n matrix with entries in K. Then AadjA = detai = adjaa. In particular, if deta 0, then A is invertible and A 1 = 1 deta adja. Proof. The result is trivial and basically pointless if n = 1, so assume n 2. Let s write down a formula for the i,jentry of AadjA: α ik C jk k=1 We first note that the diagonal entries of AadjA are all equal to deta, because if i = j in the above formula we get the Laplace expansion of deta along the i th row. If i j, then we want the result to be 0. The formula still looks a bit like the Laplace expansion of a determinant, and it is: it is the determinant of the matrix obtained from A by replacing the j th row with another copy of the i th row, so that α ik = α jk. So we are done if we can prove that the determinant of any matrix with two rows the same is zero. We proceed by induction on n. First note that the result is true for 2 2 matrices: det = αβ βα = 0. Now assume M is a n n matrix with two rows the same, and n 3. The value of the determinant only changes by a factor of 1 each time we swap rows so we may assume 33
6 that rows 2 and 3 of M are the same. Now expand detm along the first row. For each 1 j n, the 1,jminor M 1j is the determinant of an n 1 n 1 matrix with the first two rows the same, so M 1j = 0 by the induction hypothesis. Hence each cofactor C 1j = 1 1+j M 1j = 0 and therefore detm = 0. We have proved what we wanted, by induction. Adding together all these ingredients, we get that A adja has diagonal entries deta and offdiagonal entries 0, so A adja = detai. A similar proof, using expansion along columns instead of rows, works to show adjaa = detai. Now, if deta 0, then we can divide through to get A 1 deta adja 1 = deta adja A = I, so A is invertible with A 1 = 1 adja, as required. deta Corollary A is invertible if and only if deta 0. Example Let s just check this all works for 2 2 matrices. Suppose A =, γ δ with deta = αδ βγ 0. Then each minor is the determinant of a 1 1 matrix, which is easy to deal with, so we get M 11 = δ, M 12 = γ, M 21 = β and M 22 = α. Adding in the correct powers of 1, we get the cofactor matrix cofa = δ γ β α. Taking the transpose, we get adja = δ β γ α. Finally, dividing by the determinant, we get A 1 1 = αδ βγ δ β γ α, which is what we had before. 5.4 Some More Important Properties of Matrices Often in algebra, one is interested in finding the simplest representation of a particular class of objects. As the course goes on, you ll see plenty of examples of this idea for matrices, and a lot of them are based on the following concept: 34
7 Definition Let A and B be n n matrices with entries in a field K. Then we say that A is similar to B if there exists an invertible matrix C with B = CAC 1. We collect some properties of similar matrices. Lemma i Similarity is an equivalence relation on the set of n n matrices over a field K. ii Similar matrices have the same determinant. Proof. i. Since A = IAI 1 for all matrices A, A is similar to itself reflexive. If B = CAC 1, then A = C 1 BC = C 1 BC 1 1, so A is similar to B implies B is similar to A symmetric. If B 1 = C 1 AC1 1 and B 2 = C 2 B 1 C2 1, then B 2 = C 2 C 1 AC1 1 C2 1 = C 2 C 1 AC 2 C 1 1 transitive. ii. detcac 1 = detcdetadetc 1 = detcdetadetc 1 = deta. One of the important things you ll do later on in the course is work out nice representatives of the equivalence classes for the similarity relation, using the Jordan Normal Form. We finish with another property of matrices which is preserved under similarity. Definition Let A = α ij be an n n matrix with entries in K. The trace of A is the sum of the diagonal entries of A, that is tra := α ii. i=1 Lemma Let A = α ij and B = β ij be n n matrices. Then: i tra+b = tra+trb; ii trab = trba; iii Similar matrices have the same trace. Proof. i. The i,ientry of A + B is α ii + β ii, so tra + B = n i=1 α ii + β ii = n i=1 α ii+ n i=1 β ii = tra+trb. ii. The i,ientry of AB is n α ijβ ji. The i,ientry of BA is n β ijα ji. So trab = = i=1 α ij β ji β ji α ij i=1 = trba. iii. trcac 1 = trac 1 C = trai = tra, using part ii to swap the order of AC 1 and C. 35
8 Example Note that although similar matrices have the same trace and determinant, the converse is not true: the matrices A = and B have the same trace and the same determinant, but there does not exist C with B = CAC 1, because CAC 1 = A for all invertible C. Appendix to Chapter 5 This section is included for completeness, and the proofs contained here are not examinable at all. However, everyone should see a proof that the determinant is multiplicative at least once, so here goes! We begin by collecting some properties of the determinant function det : M n K K: Lemma i If two rows of A are equal, then deta = 0. ii IfthematrixB isobtainedfrom thematrixabymultiplyingonerowbyaconstant, then detb = λdeta. iii If the matrix B is obtained from the matrix A by adding one row to another, then detb = deta. Proof. We ll freely use the Laplace expansion of the determinant when we need it. i. A proof of this was given in the proof of Theorem 5.11 above. It relied on the fact that swapping two rows of a matrix simply changes the sign of the determinant. To see this recall that, by definition, deta = σ S n sgnσ n α σj,j. Swapping rows r and s, say, is equivalent to applying the transposition r, s to this formula. Since sgnr,s = 1, this just changes the sign of the determinant, as claimed. ii. Suppose the i th row of A = α jk is multiplied by λ to get B = β jk. Then, expanding along the i th row, all the cofactors for A and B are the same, so we get detb = β ij C ij = λα ij C ij = λ α ij C ij = λdeta, as required. iii. Suppose the i th row of A is added to the j th row to get the new j th row of B. Expand along this j th row to get: detb = β jk C jk = k=1 α jk +α ik C jk = k=1 α jk C jk + k=1 α ik C jk. The first term on the right hand side is deta, and the second term is the determinant of a matrix whose i th and j th rows are equal, which gives 0 by part i. So we are done. Properties ii and iii in the lemma above show us how elementary row operations affect the determinant. Next we recall that these operations can also be achieved using matrix multiplication. 36 k=1
9 Definition We define two types of elementary matrix: i For each λ K and each 1 i n, define the n n matrix R λ,i to be the diagonal matrix with a λ in the i th diagonal position and 1s in all other diagonal positions. ii For each 1 i,j n with i j, define the matrix R j i to be the n n matrix with 1s on the diagonal, a 1 in the i,jposition and 0s elsewhere. Lemma With the notation given above, for any matrix A M n K, we have: i For all λ K and 1 i n, R λ,i A is the matrix obtained from A by multiplying the i th row by λ. ii For all 1 i,j n with i j, R j ia is the matrix obtained from A by replacing the i th row with the row obtained by adding the i th and j th rows. iii We have detr λ,i A = λdeta for all λ and i and detr j ia = deta for all i,j. Proof. The statements in i and ii are easy to check. Part iii follows from Lemma 5.19ii and iii. We can now give a proof that invertible matrices have nonzero determinant which doesn t use the multiplicative property recall that our original proof in Lemma 5.6 used the multiplicative property Lemma A matrix A is invertible if and only if deta 0. Proof. Suppose deta 0. Then the proof of Theorem 5.11 still goes through without the multiplicative property, so A is invertible. Conversely, suppose A is invertible. Then, viewing A as a linear transformation from K n to K n, A is an isomorphism of vector spaces, so A has rank n. Since the rank of a matrix as a linear map equals its row rank, the rows of A must be linearly independent. Hence, by performing row operations on A which is just taking linear combinationsof therows, we can row reduceatothe identity matrix I n. In the language of Definition 5.20, we find a sequence of elementary matrices R 1,R 2,...,R r such that I n = R r R r 1 R 1 A. At each step, the determinant of the new matrix stays unchanged, or gets multiplied by some λ K. Since we end up with the identity matrix, we never multiply a row by 0 because that is an irreversible step, so we conclude from Lemma 5.21 that deti n = 1 = detr r R r 1 R 1 A = µdeta for some nonzero µ K. Hence deta = µ 1 0, as required. We finally need a result which will help us deal with the case where our matrices are not invertible. Lemma SupposeAisnotinvertible. ThenAB isnotinvertibleforanyb M n K. Proof. If B is not invertible either, then viewing B as a linear transformation on K n, we can find a nonzero vector v K n with Bv = 0, because B must have nontrivial kernel. But then ABv = A0 = 0, so AB has nontrivial kernel, so AB is not invertible. If B is invertible, with inverse B 1, choose nonzero v in the kernel of A. Then, since B 1 is invertible, we must have that u = B 1 v 0. But ABu = ABB 1 v = AI n v = Av = 0, so AB again has nontrivial kernel, so is not invertible. 37
10 We can now prove the result we want by applying the preceding lemmas. Theorem ForanytwomatricesA,B M n K, wehavedetab = detadetb. Proof. Suppose first that A is not invertible, then deta = 0, by Lemma Also, AB is not invertible by Lemma 5.23, so detab = 0 by Lemma 5.22 again. Therefore, we have 0 = detab = detadetb, as required. Now suppose A is invertible. Then, by the argument in the proof of Lemma 5.22, we can find a sequence of elementary matrices R 1,...,R r with R r R r 1 R 1 A = I n. Let R = R r R 1 ; then, also by the arguments in that Lemma, detrc = deta 1 detc for any C M n K. In particular, then, we have detb = deti n B = detrab = deta 1 detab. Multiplying through by deta, we get the result we want. Note that this proof works because we re working over a field K, so we can apply things we know about fields and vector spaces. In general, one has to be even more careful, but this is definitely beyond what we need to do now. 38
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.
More informationChapter 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
More informationLecture 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
More informationMATH36001 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
More information2.5 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 informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationMatrix Inverse and Determinants
DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and
More information2.1: 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
More informationChapter 4: Binary Operations and Relations
c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,
More information1 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
More information(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
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More information5.3 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
More informationMath 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 informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationDeterminants. 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
More information4. 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 informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationPreliminaries of linear algebra
Preliminaries of linear algebra (for the Automatic Control course) Matteo Rubagotti March 3, 2011 This note sums up the preliminary definitions and concepts of linear algebra needed for the resolution
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More information8 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 informationINTRODUCTORY 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 informationMATH 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
More informationUnit 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 informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationLecture Notes: Matrix Inverse. 1 Inverse Definition
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
More informationLinear Algebra Concepts
University of Central Florida School of Electrical Engineering Computer Science EGN3420  Engineering Analysis. Fall 2009  dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationBasics 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
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationMATRIX 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 informationMATRIX 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 informationUsing 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
More informationMath 315: Linear Algebra Solutions to Midterm Exam I
Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =
More informationSergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014
Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of
More informationPROVING STATEMENTS IN LINEAR ALGEBRA
Mathematics V2010y Linear Algebra Spring 2007 PROVING STATEMENTS IN LINEAR ALGEBRA Linear algebra is different from calculus: you cannot understand it properly without some simple proofs. Knowing statements
More informationProblems for Advanced Linear Algebra Fall 2012
Problems for Advanced Linear Algebra Fall 2012 Class will be structured around students presenting complete solutions to the problems in this handout. Please only agree to come to the board when you are
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More information9 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:
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationCofactor 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 information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More informationGRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics
GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic
More informationMatrices, transposes, and inverses
Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrixvector multiplication: two views st perspective: A x is linear combination of columns of A 2 4
More informationWe 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
More informationSolution. 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 Takehome 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 informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional 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 informationMathematics 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
More informationLecture 10: Invertible matrices. Finding the inverse of a matrix
Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationNotes on Linear Algebra. Peter J. Cameron
Notes on Linear Algebra Peter J. Cameron ii Preface Linear algebra has two aspects. Abstractly, it is the study of vector spaces over fields, and their linear maps and bilinear forms. Concretely, it is
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationThe Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices
The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created:
More informationThe Cayley Hamilton Theorem
The Cayley Hamilton Theorem Attila Máté Brooklyn College of the City University of New York March 23, 2016 Contents 1 Introduction 1 1.1 A multivariate polynomial zero on all integers is identically zero............
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationMATH 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 mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationCalculus 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.rwthaachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview
More informationSolution 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 informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More informationL12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014
L12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.  Morpheus Primary concepts:
More informationDeterminants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number
LECTURE 13 Determinants 1. Calculating the Area of a Parallelogram Definition 13.1. Let A be a matrix. [ a c b d ] The determinant of A is the number det A) = ad bc Now consider the parallelogram formed
More informationLECTURE 1 I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find x such that IA = A
LECTURE I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find such that A = y. Recall: there is a matri I such that for all R n. It follows that
More informationDISTRIBUTIVE PROPERTIES OF ADDITION OVER MULTIPLICATION OF IDEMPOTENT MATRICES
J. Appl. Math. & Informatics Vol. 29(2011), No. 56, pp. 16031608 Website: http://www.kcam.biz DISTRIBUTIVE PROPERTIES OF ADDITION OVER MULTIPLICATION OF IDEMPOTENT MATRICES WIWAT WANICHARPICHAT Abstract.
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More information4: 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 informationRow and column operations
Row and column operations It is often very useful to apply row and column operations to a matrix. Let us list what operations we re going to be using. 3 We ll illustrate these using the example matrix
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationEC9A0: Presessional Advanced Mathematics Course
University of Warwick, EC9A0: Presessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Presessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationAdditional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman
Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman Email address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents
More informationThe 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 wellknown properties Volumes of parallelepipeds are
More informationDETERMINANTS TERRY A. LORING
DETERMINANTS TERRY A. LORING 1. Determinants: a Row Operation ByProduct The determinant is best understood in terms of row operations, in my opinion. Most books start by defining the determinant via formulas
More informationHomework: 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
More informationMatrices, Determinants and Linear Systems
September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More informationAbstract Linear Algebra, Fall Solutions to Problems III
Abstract Linear Algebra, Fall 211  Solutions to Problems III 1. Let P 3 denote the real vector space of all real polynomials p(t) of degree at most 3. Consider the linear map T : P 3 P 3 given by T p(t)
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationMatrix 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
More informationDerivatives of Matrix Functions and their Norms.
Derivatives of Matrix Functions and their Norms. Priyanka Grover Indian Statistical Institute, Delhi India February 24, 2011 1 / 41 Notations H : ndimensional complex Hilbert space. L (H) : the space
More informationMath 4707: Introduction to Combinatorics and Graph Theory
Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices
More informationLecture 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 informationPractice 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
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationNON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that
NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA LINEAR ALGEBRA
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA LINEAR ALGEBRA KEITH CONRAD Our goal is to use abstract linear algebra to prove the following result, which is called the fundamental theorem of algebra. Theorem
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationThe 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 inclass presentation
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationHarvard College. Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review
Harvard College Math 21b: Linear Algebra and Differential Equations Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu May 5, 2010 1 Contents Table of Contents 4 1 Linear Equations
More informationLecture Notes 1: Matrix Algebra Part B: Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 75 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond revised 2016 September 18th University of Warwick,
More informationEconomics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics
Economics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics Michael Best Spring 2015 Outline The Evaluation Problem Matrix Algebra Matrix Calculus OLS With Matrices The Evaluation Problem:
More informationLecture 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 informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationDefinition 12 An alternating bilinear form on a vector space V is a map B : V V F such that
4 Exterior algebra 4.1 Lines and 2vectors The time has come now to develop some new linear algebra in order to handle the space of lines in a projective space P (V ). In the projective plane we have seen
More information13 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 threespace, we write a vector in terms
More information1 Gaussian Elimination
Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 GaussJordan reduction and the Reduced
More information