n×n Thm: Suppose matrix Q 2 R is orthogonal. Then −1 T I Q is invertible with Q = Q . n T T I For any x; y 2 R ,(Q x) (Q y) = x y. n I For any x 2 R , kQ xk2 = kxk2. Ex 0 1 1 1 1 1 2 2 2 2 B C B 1 1 1 1 C B − 2 2 − 2 2 C B C T H = B C ; H H = I : B C B − 1 − 1 1 1 C B 2 2 2 2 C @ A 1 1 1 1 2 − 2 − 2 2 x9.2 Orthogonal Matrices and Similarity Transformations n×n Def: A matrix Q 2 R is said to be orthogonal if its columns n (1) (2) (n)o n q ; q ; ··· ; q form an orthonormal set in R . Ex 0 1 1 1 1 1 2 2 2 2 B C B 1 1 1 1 C B − 2 2 − 2 2 C B C T H = B C ; H H = I : B C B − 1 − 1 1 1 C B 2 2 2 2 C @ A 1 1 1 1 2 − 2 − 2 2 x9.2 Orthogonal Matrices and Similarity Transformations n×n Def: A matrix Q 2 R is said to be orthogonal if its columns n (1) (2) (n)o n q ; q ; ··· ; q form an orthonormal set in R . n×n Thm: Suppose matrix Q 2 R is orthogonal. Then −1 T I Q is invertible with Q = Q . n T T I For any x; y 2 R ,(Q x) (Q y) = x y. n I For any x 2 R , kQ xk2 = kxk2. x9.2 Orthogonal Matrices and Similarity Transformations n×n Def: A matrix Q 2 R is said to be orthogonal if its columns n (1) (2) (n)o n q ; q ; ··· ; q form an orthonormal set in R . n×n Thm: Suppose matrix Q 2 R is orthogonal. Then −1 T I Q is invertible with Q = Q . n T T I For any x; y 2 R ,(Q x) (Q y) = x y. n I For any x 2 R , kQ xk2 = kxk2. Ex 0 1 1 1 1 1 2 2 2 2 B C B 1 1 1 1 C B − 2 2 − 2 2 C B C T H = B C ; H H = I : B C B − 1 − 1 1 1 C B 2 2 2 2 C @ A 1 1 1 1 2 − 2 − 2 2 Thm: Suppose A and B are similar matrices with A = S−1 BS and λ is an eigenvalue of A with associated eigenvector x. Then λ is an eigenvalue of B with associated eigenvector S x. Proof: Let x 6= 0 be such that A x = S−1 BS x = λ x: It follows that B (S x) = λ (S x) Def: Two matrices A and B are similar if a nonsingular matrix S exists with A = S−1 BS: Proof: Let x 6= 0 be such that A x = S−1 BS x = λ x: It follows that B (S x) = λ (S x) Def: Two matrices A and B are similar if a nonsingular matrix S exists with A = S−1 BS: Thm: Suppose A and B are similar matrices with A = S−1 BS and λ is an eigenvalue of A with associated eigenvector x. Then λ is an eigenvalue of B with associated eigenvector S x. Def: Two matrices A and B are similar if a nonsingular matrix S exists with A = S−1 BS: Thm: Suppose A and B are similar matrices with A = S−1 BS and λ is an eigenvalue of A with associated eigenvector x. Then λ is an eigenvalue of B with associated eigenvector S x. Proof: Let x 6= 0 be such that A x = S−1 BS x = λ x: It follows that B (S x) = λ (S x) Proof: −1 (1) (2) (n) A = SDS ; S = v ; v ; ··· ; v ; D = diag (λ1; λ2; ··· ; λn) (1) (2) (n) (1) (2) (n) () A v ; v ; ··· ; v = v ; v ; ··· ; v diag (λ1; λ2; ··· ; λn) ; (j) (j) (1) (2) (n) () A v = λj v ; j = 1; 2; ··· ; n: v ; v ; ··· ; v L.I.D. n Cor: A 2 R with n distinct eigenvalues is similar to diagonal matrix. n Thm: A 2 R is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. n Cor: A 2 R with n distinct eigenvalues is similar to diagonal matrix. n Thm: A 2 R is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. Proof: −1 (1) (2) (n) A = SDS ; S = v ; v ; ··· ; v ; D = diag (λ1; λ2; ··· ; λn) (1) (2) (n) (1) (2) (n) () A v ; v ; ··· ; v = v ; v ; ··· ; v diag (λ1; λ2; ··· ; λn) ; (j) (j) (1) (2) (n) () A v = λj v ; j = 1; 2; ··· ; n: v ; v ; ··· ; v L.I.D. n Thm: A 2 R is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. Proof: −1 (1) (2) (n) A = SDS ; S = v ; v ; ··· ; v ; D = diag (λ1; λ2; ··· ; λn) (1) (2) (n) (1) (2) (n) () A v ; v ; ··· ; v = v ; v ; ··· ; v diag (λ1; λ2; ··· ; λn) ; (j) (j) (1) (2) (n) () A v = λj v ; j = 1; 2; ··· ; n: v ; v ; ··· ; v L.I.D. n Cor: A 2 R with n distinct eigenvalues is similar to diagonal matrix. n Schur Thm: Let A 2 R .A unitary matrix U exists such that 0 1 B C −1 B C T = U AU = B C is upper-triangular. @ A The diagonal entries of T are the eigenvalues of A. n×n Def: A matrix U 2 C is unitary if kU xk2 = kxk2 for any vector x. n×n Def: A matrix U 2 C is unitary if kU xk2 = kxk2 for any vector x. n Schur Thm: Let A 2 R .A unitary matrix U exists such that 0 1 B C −1 B C T = U AU = B C is upper-triangular. @ A The diagonal entries of T are the eigenvalues of A. p p T T T Def: kuk2 = a a + b b = u u. T n Thm: Let A = A 2 R be symmetric. Then all eigenvalues of A are real. Proof: Let λ be an eigenvalue of A with eigenvector u. Then λ is eigenvalue, A u = λ u; −! A u = λ u: λ uT u = uT (A u) = (A u)T u = λ uT u : Therefore λ = λ 2 R: Def: The complexp conjugate of ap complex vector u = a + −1 b 2 Cn is u = a − −1 b. T n Thm: Let A = A 2 R be symmetric. Then all eigenvalues of A are real. Proof: Let λ be an eigenvalue of A with eigenvector u. Then λ is eigenvalue, A u = λ u; −! A u = λ u: λ uT u = uT (A u) = (A u)T u = λ uT u : Therefore λ = λ 2 R: Def: The complexp conjugate of ap complex vector u = a + −1 b 2 Cn is u = a − −1 b. p p T T T Def: kuk2 = a a + b b = u u. Proof: Let λ be an eigenvalue of A with eigenvector u. Then λ is eigenvalue, A u = λ u; −! A u = λ u: λ uT u = uT (A u) = (A u)T u = λ uT u : Therefore λ = λ 2 R: Def: The complexp conjugate of ap complex vector u = a + −1 b 2 Cn is u = a − −1 b. p p T T T Def: kuk2 = a a + b b = u u. T n Thm: Let A = A 2 R be symmetric. Then all eigenvalues of A are real. Def: The complexp conjugate of ap complex vector u = a + −1 b 2 Cn is u = a − −1 b. p p T T T Def: kuk2 = a a + b b = u u. T n Thm: Let A = A 2 R be symmetric. Then all eigenvalues of A are real. Proof: Let λ be an eigenvalue of A with eigenvector u. Then λ is eigenvalue, A u = λ u; −! A u = λ u: λ uT u = uT (A u) = (A u)T u = λ uT u : Therefore λ = λ 2 R: Proof: I By induction on n. Assume theorem true for n − 1. I Let λ be eigenvalue of A with unit eigenvector u: A u = λ u. n I We extend u into an orthonormal basis for R : u; u2; ··· ; un are unit, mutually orthogonal vectors. def def n×n I U = (u; u2; ··· ; un) = u; Ub 2 R is orthogonal. 0 1 T uT (A u) uT A U T u b U AU = T A u; A Ub = @ A Ub UbT (A u) UbT A Ub λ 0T ! = : 0 UbT A Ub T (n−1)×(n−1) I Matrix Ub A Ub 2 R is symmetric. n Thm: A matrix A 2 R is symmetric if and only if there exists a n diagonal matrix D 2 R and an orthogonal matrix Q so that 0 1 B C T B C T A = QDQ = Q B C Q . @ A n Thm: A matrix A 2 R is symmetric if and only if there exists a n diagonal matrix D 2 R and an orthogonal matrix Q so that 0 1 B C T B C T A = QDQ = Q B C Q . @ A Proof: I By induction on n.