Homework Answers
2. Let A be an n x n real symmetric matrix or a complex normal matrix....
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
Help on Questions 1-3 Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x...
Help on Questions 1-3
Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...
True or False? (a) An n x n matrix that is diagonalizable must be symmetric. (b)...
True or False? (a) An n x n matrix that is diagonalizable must be symmetric. (b) If AT = A and if vectors u and v satisfy Au = 3u and Av = 40, then u: v=0. (c) An n x n symmetric matrix has n distinct real eigenvalues. (d) For a nonzero v in R", the matrix vv7 is a rank-1 matrix.
Need help with linear algebra problem! Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors o...
Need help with linear algebra problem!
Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that if v E R2 is a vector such that û1)Su = 0, then 5 = Bû(2) for some B 0.
Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that...
Let A be an n × n real symmetric matrix with its row and column sums both equal to 0. Let λ1, . . . , λn be the eigenvalues of A, with λn = 0, and with corresponding eigenvectors v1,...,vn (these exis...
Let A be an n × n real symmetric matrix with its row and column sums both equal to 0. Let λ1, . . . , λn be the eigenvalues of A, with λn = 0, and with corresponding eigenvectors v1,...,vn (these exist because A is real symmetric). Note that vn = (1, . . . , 1). Let A[i] be the result of deleting the ith row and column. Prove that detA[i] = (λ1···λn-1)/n. Thus, the number of spanning...
Suppose A is a symmetric n x n matrix with n positive eigenvalues. Explain why an...
Suppose A is a symmetric n x n matrix with n positive eigenvalues. Explain why an orthogonal diagonalization A = PDPT of A is also a singular value decomposition of A, with U = P =V and E = D. [Hint: First, explain why this is equivalent to showing the singular values of A are exactly the eigenvalues of A. Then show this is the case with these assumptions on A.]
Here is Gerschgorin's theorem, which holds for any m x m matrix A, symmetric or nonsym- metric. E...
Here is Gerschgorin's theorem, which holds for any m x m matrix A, symmetric or nonsym- metric. Every eigenvalue of A lies in at least one of the m circular disks in the complex plane with centers au and radii Σ (41. Moreover, if n of these disks forrn a connected domain that is disjoint from the other m-n disks, then there are precisely n eigenvalues of A within this domain. Prove the first part of Gerschgorin's theorem. (Hint: Let...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P su...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
3, (a) [5 marks] what does it mean for A E Rnxn to be (i) symmetric?...
3, (a) [5 marks] what does it mean for A E Rnxn to be (i) symmetric? (ii) orthogonal? (ii) diagonalisable? (iv) orthogonally diagonalisable? (b) [4 marks] Suppose that A ERn is orthogonally diagonalisable. Prove that A is symmetric. (c) [11 marks] Let A be the matrix 6 -2 Show that the eigenvalues are 7 and -6. Show that any corresponding eigenvectors vi and v2 are orthogonal with respect to the Euclidean inner product (d) [5 marks] Hence prove that the...
estion 3 Let A be an n x n symmetric matrix. Then, which of the following...
estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
Help on Questions 1-3
Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...
True or False? (a) An n x n matrix that is diagonalizable must be symmetric. (b) If AT = A and if vectors u and v satisfy Au = 3u and Av = 40, then u: v=0. (c) An n x n symmetric matrix has n distinct real eigenvalues. (d) For a nonzero v in R", the matrix vv7 is a rank-1 matrix.
Need help with linear algebra problem!
Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that if v E R2 is a vector such that û1)Su = 0, then 5 = Bû(2) for some B 0.
Let S be a symmetric, 2 x 2 matrix. Let û1) and ût2) be orthogonal eigenvectors of S with corresponding nonzero eigenvalues A1 and X2. Show that...
Suppose A is a symmetric n x n matrix with n positive eigenvalues. Explain why an orthogonal diagonalization A = PDPT of A is also a singular value decomposition of A, with U = P =V and E = D. [Hint: First, explain why this is equivalent to showing the singular values of A are exactly the eigenvalues of A. Then show this is the case with these assumptions on A.]
Here is Gerschgorin's theorem, which holds for any m x m matrix A, symmetric or nonsym- metric. Every eigenvalue of A lies in at least one of the m circular disks in the complex plane with centers au and radii Σ (41. Moreover, if n of these disks forrn a connected domain that is disjoint from the other m-n disks, then there are precisely n eigenvalues of A within this domain. Prove the first part of Gerschgorin's theorem. (Hint: Let...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
3, (a) [5 marks] what does it mean for A E Rnxn to be (i) symmetric? (ii) orthogonal? (ii) diagonalisable? (iv) orthogonally diagonalisable? (b) [4 marks] Suppose that A ERn is orthogonally diagonalisable. Prove that A is symmetric. (c) [11 marks] Let A be the matrix 6 -2 Show that the eigenvalues are 7 and -6. Show that any corresponding eigenvectors vi and v2 are orthogonal with respect to the Euclidean inner product (d) [5 marks] Hence prove that the...
estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.
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