7. (10) Find all n xn orthogonal, symmetric, and positive definite real matrix (matrices). Explain your...
linear algebra
Find all n x n orthogonal, symmetric, and positive definite real matrix (matrices). Explain answer
true or false and explain why
(a) If the eigenvalues of a real symmetric matrix Anxn are all positive, then 7" A7 > 0 for any i in R" (b) If a real square matrix is orthogonally diagonalizable, it must be symmetric. (c) If A is a real mx n matrix, then both APA and AA' are semi-positive definite. (d) SVD and orthogonal diagonalization coincide when the real matrix concerned is symmetric pos- itive definite. (e) If vectors and q...
3. Answer the following questions regarding positive definite matrix. A symmetric real matrix M is said to be positive definite if the scalar 27 Mz is positive for every non-zero column vector z (a) Consider the matrix [9 6] A = 6 a so that the matrix A is positive definite? What should a satisfy (b) Suppose we know matrix B is positive definite. Show that B1 is also positive definite. Hint use the definition and the fact that every...
(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...
1. For each of the following symmetric matrices, find an orthogonal matrix P and diagonal matrix D such that PTAP = D. 0 1 (а) А — 1 0 1 -1 1 0 2 -2 (Ъ) А %— -2 -2 -4 -2 2 |3 0 7 0 5 0 7 0 3 (с) А %—
1. For each of the following symmetric matrices, find an orthogonal matrix P and diagonal matrix D such that PTAP = D. 0 1 (а)...
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Prove that this function defines a vector norm. Hint: Use the Cholesky decomposition. (b) Find an example of square matrices A an This shows that ρ(A) is not a norm. Note: there are very simple examples. d B such that ρ(A+B)>ρ(A) + ρ(8)
(a) Let S be a symmetric positive definite matrix and define a function | on R"...
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 +...
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.]
A. (For Math 603 students only) Consider a symmetric and positive definite matrix A Rnxn and let λ'nin(A) and Xmax(A) be the minimal and maximal real eigenvalues of A respectively. Show that Suggested readings: Sections 7.2, 7.5, 7.6
A. (For Math 603 students only) Consider a symmetric and positive definite matrix A Rnxn and let λ'nin(A) and Xmax(A) be the minimal and maximal real eigenvalues of A respectively. Show that Suggested readings: Sections 7.2, 7.5, 7.6
2. Suppose that is symmetric, positive definite and A is the lower triangular matrix given by the Cholesky factorization. Prove that, if X N (0,) then Y = AXN (E).