2) If A is a symmetric matrix, show that the norm of A can be the...
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix
4. Show that an arbitrary square matrix A can be written as where Ai is a symmetric matrix and A2 is a skew-symmetric matrix
Consider the singular value decomposition (svd) of a symmetric matrix, A- UAU Show that for any integer, n, An-UNU. Argue that for a psd matrix A, there must exist a square root matrix, A-such that 1/2 1/2 A 1/2
QUESTION 6 (2 pts). Exercise 2.3.2 Suppose A є Mn,n(F) and that λ is an eigenvalue of A. Show that, for any choice of vector norm on Fn, we have lAll-A, where |All is the associated matrix norm of A.
QUESTION 6 (2 pts). Exercise 2.3.2 Suppose A є Mn,n(F) and that λ is an eigenvalue of A. Show that, for any choice of vector norm on Fn, we have lAll-A, where |All is the associated matrix norm of A.
2. Let A € Mn(R). (a) Show that AAT is a semipositive definite symmetric matrix and that AAT and AT A are similar. (b) Show by example that it need not be the case that AAT and ATA are similar for A E Mn(C).
d) Let it-| 1 | and v2 = 101 be eigenvectors for a symmetric matrix A. If the eigenvalue of is 2, what is the eigenvalue of matrix that diagonalises the matrix A Find an orthogonal
d) Let it-| 1 | and v2 = 101 be eigenvectors for a symmetric matrix A. If the eigenvalue of is 2, what is the eigenvalue of matrix that diagonalises the matrix A Find an orthogonal
(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"...
The symmetric matrix A below has distinct eigenvalues 10,-2 and-8. Find an orthogonal matrix P and a diagonal matrix D such that pTAP-Duse the square root symbol 'where needed to give an exact value for your answer. -1 47 A- 4 2-4 0 0 0] P=10 0 0| D=10 0 0
(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...
Assume a symmetric matrix 3 x 3 matrix has eigenvalues 2 and 3 and that the eigenspace of 2 is E2 = span -{80 Which of the following is an eigenvector for the eigenvalue 3? O -6 O Por O 37
υΣνΤ. Answer the following questions: Suppose a matrix A E Rmxn has an SVD A (i) Show that the rank of the miatrix A E Rmxn is equal to the number of its nonzero singular values. (ii) Show that miultiplication by an orthogonal matrix on the left and multiplication by an orthogonal matrix on the right, i.e., UA and BU, where A E Rmxn and B ERnm are general matrices, and U Rxm is an orthogonal matrix, preserve the Frobenius...