# 4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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: Notation 8 1. Review Lemma 1.1. Let V be a finite-dimensional vector space over a field F. Let β, β be two bases for V. Let T : V V be a linear transformation. Define Q := [I V ] β β. Then [T ]β β and [T ] β β satisfy the following relation [T ] β β = Q[T ] β β Q 1. Theorem 1.2. Let A be an n n matrix. Then A is invertible if and only if det(a) 0. Exercise 1.3. Let A be an n n matrix with entries A ij, i, j {1,..., n}, and let S n denote the set of all permutations on n elements. For σ S n, let sign(σ) := ( 1) N, where σ can be written as a composition of N transpositions. Then det(a) = n sign(σ) A iσ(i). σ S n 2. Diagonal Matrices So far, we should have a reasonably good understanding of linear transformations, matrices, rank and invertibility. However, given a matrix, we don t yet have a good understanding of how to simplify this matrix. In mathematics and science, the general goal is to take some complicated and make it simpler. In the context of linear algebra, this paradigm becomes: try to find a particular basis such that a linear transformation has a diagonal matrix representation. (After all, diagonal matrices are among the simplest matrices.) We now attempt to realize this goal within our discussion of eigenvectors and diagonalization. Definition 2.1 (Diagonal Matrix). An n n matrix A with entries A ij, i, j {1,..., n} is said to be diagonal if A ij = 0 whenever i j, i, j {1,..., n}. If A is diagonal, we denote the matrix A by diag(a 11, A 22,..., A nn ). Date: November 19,

2 Lemma 2.2. The rank of a diagonal matrix is equal to the number of its nonzero entries. 3. Eigenvectors and Eigenvalues Definition 3.1 (Eigenvector and Eigenvalue). Let V be a vector space over a field F. Let T : V V be a linear transformation. An eigenvector of T is a nonzero vector v V such that, there exists λ F with T (v) = λv. The scalar λ is then referred to as the eigenvalue of the eigenvector v. The equation T (v) = λv is self- Remark 3.2. The word eigen is German for self. referential in v, which explains the etymology here. Example 3.3. If A is diagonal with A = diag(a 11,..., A nn ), then L A has eigenvectors (e 1,..., e n ) with eigenvalues (A 11,..., A nn ). Example 3.4. If T is the identity transformation, then every vector is an eigenvector with eigenvalue 1. Example 3.5. If T : V V has v N(T ) with v 0, then v is an eigenvector of T with eigenvalue zero. Example 3.6. Define T : C (R) C (R) by T (f) := f. For any y R, the function f(x) := e ixy satisfies T f(x) = f (x) = y 2 f(x). So, for any y R, e ixy is an eigenfunction of T with eigenvalue y 2. Definition 3.7 (Eigenspace). Let V be a vector space over a field F. Let T : V V be a linear transformation. Let λ F. The eigenspace of λ is the set of all v V (including zero) such that T (v) = λv. Remark 3.8. Given λ F, the set of v such that T (v) = λv is the same as N(T λi V ). In particular, an eigenspace is a subspace of V. And N(T λi V ) is nonzero if and only if T λi V is not one-to-one. Lemma 3.9 (An Eigenvector Basis Diagonalizes T ). Let V be an n-dimensional vector space over a field F, and let T : V V be a linear transformation. Suppose V has an ordered basis β := (v 1,..., v n ). Then v i is an eigenvector of T with eigenvalue λ i F, for all i {1,..., n}, if and only if the matrix [T ] β β is diagonal with [T ]β β = diag(λ 1,..., λ n ). Proof. We begin with the forward implication. Let i {1,..., n}. Suppose T (v i ) = λ i v i, [T (v i )] β is a column vector whose i th entry is λ i, with all other entries zero. Since [T ] β β = ([T (v 1 )] β,..., [T (v n )] β ), we conclude that [T ] β β = diag(λ 1,..., λ n ). Conversely, suppose [T ] β β = diag(λ 1,..., λ n ). Since [T ] β β = ([T (v 1)] β,..., [T (v n )] β ), we conclude that T (v i ) = λ i v i for all i {1,..., n}, so that v i is an eigenvector of T with eigenvalue λ i, for all i {1,..., n}. Definition 3.10 (Diagonalizable). A linear transformation T : V V is said to be diagonalizable if there exists an ordered basis β of V such the matrix [T ] β β is diagonal. Remark From Lemma 3.9, T is diagonalizable if and only if it has a basis consisting of eigenvectors of T. 2

3 Example Let T : R 2 R 2 denote reflection across the line l which passes through the origin and (1, 2). Then T (1, 2) = (1, 2), and T (2, 1) = (2, 1), so we have two eigenvectors of T with eigenvalues 1 and 1 respectively. The vectors ((1, 2), (2, 1)) are independent, so they form a basis of R 2. From Lemma 3.9, T is diagonalizable. For β := ((1, 2), (2, 1)), we have [T ] β 1 0 β = diag(1, 1) =. 0 1 Note that [T 2 ] β β = I 2 = [I R 2] β β, so T 2 = I R 2. The point of this example is that, once we can diagonalize T, taking powers of T becomes very easy. Definition 3.13 (Diagonalizable Matrix). An n n matrix A is diagonalizable if the corresponding linear transformation L A is diagonalizable. Lemma A matrix A is diagonalizable if and only if there exists an invertible matrix Q and a diagonal matrix D such that A = QDQ 1. That is, a matrix A is diagonalizable if and only if it is similar to a diagonal matrix. Proof. Suppose A is an n n diagonalizable matrix. Let β denote the standard basis of F n, so that A = [L A ] β β. Since A is diagonalizable, there exists an ordered basis β such that D := [L A ] β β is diagonal. From Lemma 1.1, there exists an invertible matrix Q := [I F n] β β such that A = [L A ] β β = Q[L A] β β Q 1 = QDQ 1. We now prove the converse. Suppose A = QDQ 1, where Q is invertible and D is diagonal. Let λ 1,..., λ n such that D = diag(λ 1,..., λ n ). Then De i = λ i e i for all i {1,..., n}, so A(Qe i ) = QDQ 1 Qe i = QDe i = λ i Qe i. So, Qe i is an eigenvector of A, for each i {1,..., n}. Since Q is invertible and (e 1,..., e n ) is a basis of F n, we see that (Qe 1,..., Qe n ) is also a basis of F n. So, β := (Qe 1,..., Qe n ) is a basis of F n consisting of eigenvectors of A, so A is diagonalizable by Lemma 3.9, since [L A ] β β is diagonal. Lemma Let A be an n n matrix. Suppose β = (v 1,..., v n ) is an ordered basis of F n such that v i is an eigenvector of A with eigenvalue λ i for all i {1,..., n}. Let Q be the matrix with columns v 1,..., v n (where we write each v i in the standard basis). Then A = Q diag(λ 1,..., λ n )Q 1. Proof. Let β be the standard basis of F n. Note that [I F n] β β = Q. So, by Lemma 1.1, Since L A v i = λ i v i for all i {1,..., n}, A = [L A ] β β = Q[L A] β β Q 1. The Lemma follows. [L A ] β β = diag(λ 1,..., λ n ). 3

4 4. Characteristic Polynomial Lemma 4.1. Let A be an n n matrix. Then λ F is an eigenvalue of A if and only if det(a λi n ) = 0. Proof. Suppose λ is an eigenvalue of A. Then there exists v F n such that Av = λv and v 0, so that (A λi n )v = 0. So, (A λi n ) is not invertible, and det(a λi n ) = 0, from the contrapositive of Theorem 1.2. Conversely, if det(a λi n ) = 0, then A λi n is not invertible, from the contrapositive of Theorem 1.2. In particular, A λi n is not one-to-one. So, there exists v F n with v 0 such that (A λi n )v = 0, i.e. Av = λv. Definition 4.2 (Characteristic Polynomial). Let A be an n n with entries in a field F. Let λ F, and define the characteristic polynomial f(λ) of A, by f(λ) := det(a λi n ). Lemma 4.3. Let A, B be similar matrices. Then A, B have the same characteristic polynomial. Proof. Let λ F. Since A, B are similar, there exists an invertible matrix Q such that A = QBQ 1. So, using the multiplicative property of the determinant, det(a λi) = det(qbq 1 λi) = det(q(b λi)q 1 ) = det(q) det(b λi) det(q 1 ) = det(q) det(q) 1 det(b λi) = det(b λi). Lemma 4.4. Let A be an n n matrix all of whose entries lie in P 1 (F). Then det(a) P n (F). Proof. From Exercise 1.3 from the homework, det(a) is a sum of polynomials of degree at most n. That is, det(a) itself is in P n (R). Remark 4.5. From this Lemma, we see that the characteristic polynomial of A is a polynomial of degree at most n. Lemma 4.6. Let A be an n n matrix with entries A ij, i, j {1,..., n}. Then there exists g P n 2 (F) such that f(λ) = det(a λi) = (A 11 λ) (A nn λ) + g(λ) Proof. Let B := A λi. From Exercise 1.3 from the homework, n det(a λi) = (A ii λ) + sign(σ) σ S n : σ I n n B iσ(i). Note that each term in the sum on the right has a number of λ terms equal to the number of i {1,..., n} such that i = σ(i). So, if σ S n and σ I n, it suffices to show that there exist at least two integers i, j {1,..., n} with i j such that σ(i) i and σ(j) j. We prove this assertion by contradiction. Suppose there exists σ S n, σ I n with exactly one i {1,..., n} with σ(i) i. Then σ(k) = k for all k {1,..., n} {i}. Since σ is a permutation, σ is onto, so there exists i {1,..., n} such that σ(i ) = i. Since σ(k) = k 4

5 for all k {1,..., n} {i}, we must therefore have i = i, so that σ(i) = i, a contradiction. We conclude that since σ I n, there exist at least two i, j {1,..., n} with i j such that σ(i) i and σ(j) j, as desired. Definition 4.7 (Trace). Let A be an n n matrix with entries A ij, i, j {1,..., n}. Then the trace of A, denoted by Tr(A), is defined as n Tr(A) := A ii. Theorem 4.8. Let A be an n n matrix. There exist scalars a 1,..., a n 2 F such that the characteristic polynomial f(λ) of A satisfies f(λ) = ( 1) n λ n + ( 1) n 1 Tr(A)λ n 1 + a n 2 λ n a 1 λ + det(a). Proof. From Lemma 4.6, there exists g P n 2 (F) such that f(λ) = (A 11 λ) (A nn λ) + g(λ). Multiplying out the product terms, we therefore get the two highest order terms of f. That is, there exists G P n 2 (F) such that f(λ) = ( λ) n + Tr(A)( λ) n 1 + G(λ). Finally, to get the zeroth order term of the polynomial f, note that by definition of the characteristic polynomial, f(0) = det(a). Example 4.9. Let a, b, c, d R. Then the characteristic polynomial of a b c d is (a λ)(d λ) bc = λ 2 λ(a + d) + (ad bc) = λ 2 λtr(a) + det(a). Example The characteristic polynomial of is λ 2 1 = (λ + 1)(λ 1). Example Let i := 1. The characteristic polynomial of is λ = (λ + i)(λ i). However, we cannot factor λ using only real numbers. So, as we will see below, we can diagonalize this matrix over the complex numbers, but not over the real numbers. Theorem 4.12 (The Fundamental Theorem of Algebra). Let f(λ) be a real polynomial of degree n. Then there exist λ 0, λ 1,... λ n C such that f(λ) = λ 0 n (λ λ i ). 5

6 Remark This theorem is one of the reasons that complex numbers are useful. If we have complex numbers, then any real matrix has a characteristic polynomial that can be factored into complex roots. Without complex numbers, we could not do this. 5. Diagonalizability Recall that n linearly independent vectors in F n form a basis of F n. So, from Lemma 3.9 or the proof of Lemma 3.14, we have Lemma 5.1. Let A be an n n matrix with elements in F. Then A is diagonalizable (over F) if and only if there exists a set (v 1,..., v n ) of linearly independent vectors in F n such that v i is an eigenvector of A for all i {1,..., n}. We now examine some ways of finding a set of linearly independent eigenvectors of A, since this will allow us to diagonalize A. Proposition 5.2. Let A be an n n matrix. Let v 1,..., v k be eigenvectors of A with eigenvalues λ 1,..., λ k, respectively. If λ 1,..., λ k are all distinct, then the vectors v 1,..., v k are linearly independent. Proof. We argue by contradiction. Assume there exist α 1,..., α k F not all zero such that k α i v i = 0. Without loss of generality, α 1 0. Applying (A λ k I) to both sides, k 1 k 1 0 = α i (A λ k I)v i = α i (λ i λ k )v i. We now apply (A λ k 1 I) to both sides, and so on. Continuing in this way, we eventually get the equality 0 = α 1 (λ 1 λ k )(λ 1 λ k 1 ) (λ 1 λ 2 )v 1. Since λ 1,..., λ k are all distinct, and α 1 0, and since v 1 0 (since it is an eigenvector), we have arrived at a contradiction. We conclude that v 1,..., v k are linearly independent. Corollary 5.3. Let A be an n n matrix with elements in F. Suppose the characteristic polynomial f(λ) of A can be written as f(λ) = n (λ i λ), where λ i F are distinct, for all i {1,..., n}. Then A is diagonalizable. Proof. For all i {1,..., n}, let v i F n be the eigenvector corresponding to the eigenvalue λ i. Setting k = n in Proposition 5.2 shows that v 1,..., v n are linearly independent. Lemma 5.1 therefore completes the proof. Example 5.4. Consider The characteristic polynomial is then A = f(λ) = (1 λ)(4 λ) + 2 = λ 2 5λ + 6 = (λ 2)(λ 3). 6

7 So, f(λ) has two distinct real roots, and we can diagonalize A over R. Observe that v 1 = (2, 1) is an eigenvector with eigenvalue 2 and v 2 = (1, 1) is an eigenvector with eigenvalue 3. So, if we think of the eigenvectors as column vectors, and use them to define Q, 2 1 Q :=, 1 1 we then have the desired diagonalization = Q Q = Exercise 5.5. Using the matrix from Example 4.11, find its diagonalization over C. In summary, if A is an n n matrix with elements in F, and if we can write the characteristic polynomial of A as a product of n distinct roots in F, then A is diagonalizable over F. On the other hand, if we cannot write the characteristic polynomial as a product of n roots in F, then A is not diagonalizable over F. (Combining Lemmas 3.14 and 4.3 shows that, if A is diagonalizable, then it has the same characteristic polynomial as a diagonal matrix. That is, the characteristic polynomial of A is the product of n roots.) (Recalling Example 4.11, the real matrix with characteristic polynomial λ can be diagonalized over C but not over R.) The only remaining case to consider is when the characteristic polynomial of A can be written as a product of n non-distinct roots of F. Unfortunately, this case is more complicated. It can be dealt with, but we don t have to time to cover the entire topic. The two relevant concepts here would be the Jordan normal form and the minimal polynomial. To see the difficulty, note that the matrix is diagonal, so it is diagonalizable. Also, the standard basis of R 2 are eigenvectors, and the characteristic polynomial is (2 λ) 2. On the other hand, consider the matrix 2 1 A =. 0 2 This matrix also has characteristic polynomial (2 λ) 2, but it is not diagonalizable. To see this, we will observe that the eigenvectors of A do not form a basis of R 2. Since 2 is the only eigenvalue, all of the eigenvectors are in the null space of However, this matrix has only a one-dimensional null space, which is spanned by the column vector (1, 0). Since the eigenvectors of A do not form a basis of R 2, A is not diagonalizable, by Remark 3.11 (or Lemma 3.9). 7

8 6. Appendix: Notation Let A, B be sets in a space X. Let m, n be a nonnegative integers. Let F be a field. Z := {..., 3, 2, 1, 0, 1, 2, 3,...}, the integers N := {0, 1, 2, 3, 4, 5,...}, the natural numbers Q := {m/n: m, n Z, n 0}, the rationals R denotes the set of real numbers C := {x + y 1 : x, y R}, the complex numbers denotes the empty set, the set consisting of zero elements means is an element of. For example, 2 Z is read as 2 is an element of Z. means for all means there exists F n := {(x 1,..., x n ): x i F, i {1,..., n}} A B means a A, we have a B, so A is contained in B A B := {x A: x / B} A c := X A, the complement of A A B denotes the intersection of A and B A B denotes the union of A and B C(R) denotes the set of all continuous functions from R to R P n (R) denotes the set of all real polynomials in one real variable of degree at most n P (R) denotes the set of all real polynomials in one real variable M m n (F) denotes the vector space of m n matrices over the field F I n denotes the n n identity matrix det denotes the determinant function S n denotes the set of permutations on {1,..., n} sign(σ) := ( 1) N where σ S n can be written as the composition of N transpositions Tr denotes the trace function 6.1. Set Theory. Let V, W be sets, and let f : V W be a function. Let X V, Y W. f(x) := {f(v): v V }. f 1 (Y ) := {v V : f(v) Y }. The function f : V W is said to be injective (or one-to-one) if: for every v, v V, if f(v) = f(v ), then v = v. The function f : V W is said to be surjective (or onto) if: for every w W, there exists v V such that f(v) = w. 8

9 The function f : V W is said to be bijective (or a one-to-one correspondence) if: for every w W, there exists exactly one v V such that f(v) = w. A function f : V W is bijective if and only if it is both injective and surjective. Two sets X, Y are said to have the same cardinality if there exists a bijection from V onto W. The identity map I : X X is defined by I(x) = x for all x X. To emphasize that the domain and range are both X, we sometimes write I X for the identity map on X. Let V, W be vector spaces over a field F. Then L(V, W ) denotes the set of linear transformations from V to W, and L(V ) denotes the set of linear transformations from V to V. UCLA Department of Mathematics, Los Angeles, CA address: 9

### 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)(

### 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

### 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

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

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

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

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

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

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### University 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

### 1 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

### 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

### 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

### Problems 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

### 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 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

### LINEAR ALGEBRA W W L CHEN

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

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### 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

### 1 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

### 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

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

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

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

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

### MAT 242 Test 2 SOLUTIONS, FORM T

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

### 7 - Linear Transformations

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

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### 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

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### 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.

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### 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

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

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

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Notes on Symmetric Matrices

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

### 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 LINEAR ALGEBRA 1

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

### 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

### 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

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### Sergei 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

### 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

### 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

### 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

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

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

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

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

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

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

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### 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 + +

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Tv = Tu 1 + + Tu N. (u 1,...,u M, v 1,...,v N ) T(Sv) = S(Tv) = Sλv = λ(sv).

54 CHAPTER 5. EIGENVALUES AND EIGENVECTORS 5.6 Solutions 5.1 Suppose T L(V). Prove that if U 1,...,U M are subspaces of V invariant under T, then U 1 + + U M is invariant under T. Suppose v = u 1 + + u

### Topic 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

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

### 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

### Chapter 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

### The 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

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

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

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

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

### 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)

### Additional 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 E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

### 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,

### 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

### LINEAR ALGEBRA. September 23, 2010

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

### Linear Least Squares

Linear Least Squares Suppose we are given a set of data points {(x i,f i )}, i = 1,...,n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. One

### (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

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

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

### Presentation 3: Eigenvalues and Eigenvectors of a Matrix

Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues

### Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013

Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,

### 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

### x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3

Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### 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

### Methods for Finding Bases

Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### 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

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

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

### Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

### 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

### LINEAR ALGEBRA AND MATRICES M3P9

LINEAR ALGEBRA AND MATRICES M3P9 December 11, 2000 PART ONE. LINEAR TRANSFORMATIONS Let k be a field. Usually, k will be the field of complex numbers C, but it can also be any other field, e.g., the field

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

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

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### 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

### Applied Linear Algebra I Review page 1

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

### 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)

### Definition 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 2-vectors 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

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### 2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

### 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

### Section 7.4 Matrix Representations of Linear Operators

Section 7.4 Matrix Representations of Linear Operators Definition. Φ B : V à R n defined as Φ B (c 1 v 1 +c v + + c n v n ) = [c 1 c c n ] T. Property: [u + v] B = [u] B + [v] B and [cu] B = c[u] B for

### 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

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### Notes on Jordan Canonical Form

Notes on Jordan Canonical Form Eric Klavins University of Washington 8 Jordan blocks and Jordan form A Jordan Block of size m and value λ is a matrix J m (λ) having the value λ repeated along the main

### 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

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### 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

### Lecture 1: Schur s Unitary Triangularization Theorem

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