Let A, B,C be matrices with the singular value decompositions 1. A-(4/5-3/5) ( 0 0 1 0 2 0 0 0 10...
Homework problem: Singular Value Decomposition Let A E R n 2 mn. Consider the singular value decomposition A = UEVT. Let u , un), v(1),...,v(m), and oi,... ,ar denote the columns of U, the columns of V and the non-zero entries (the singular values) of E, respectively. Show that 1. ai,.,a are the nonzero eigenvalues of AAT and ATA, v(1)... , v(m) the eigenvectors of ATA and u1)...,un) (possibly corresponding to the eigenvalue 0) are the eigenvectors of AAT are...
5. Let A 2 Rm£n. Show that (a) kerA = kerAtA; (b) rankAtA = rankAAt = rankA; (c) AtA and AAt have the same nonzero eigenvalues. Hint: Keep in mind the Singular Value Decomposition of matrices.
2 0 -21 3. Let A= 1 3 2 LO 0 3 (a) Find the characteristic equation of A. in Find the other (b) One of the eigenvalues for A is ) = 2 with corresponding eigenvector 1 10 eigenvalue and a basis for the eigenspace associated to it. (e) Find matrices S and B that diagonalize A, if possible.
6. (20') Given the 3 x 3 matrix A= 0 0 1 0 2 0 4 0 0 (a) compute ATA. (b) find all eigenvalues of ATA and their associated eigenvectors. (c) write down all singular values of A in descending order. (d) find the singular-value decomposition(SVD) A = UEVT. (e) based on the above calculation, write down the SVD for the following matrix B. (You can certainly perform all the work again if you have sufficient time but do...
(4) The following is the singular value decomposition of a 3 x 4 matrix A with some entries not given 1/3 -2/V5 1/v5 2/3 2/3 3 0 12/13 5/13 3/5 4/5 5/13 12/13 0 0 A 0 2 0 0 0 0 0 0 0 (a) What are the eigenvalues of AAT? of ATA? What is the rank of A? 1 2 (b) Find a non-zero vector w such that AAT = 9w. such that ATAu 4u. (c) Find a...
8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P = -1 1 We know that A and B have the same characteristic polynomial and the same eigenvalues. Suppose that 2 is one of the common eigenvalues and x = [4 1] is a corresponding eigenvector of A. Which of the following is the eigenvector of B corresponding to 2 ? "f1-4-0-0-0-0-01-10 #8: Select
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
1. 4 2 0 A-1 1 1 0 0 3 (a) Find the characteristic polynomial of A. (b) What are A's eigenvalues? (c) Find the corresponding eigenvectors (d) Is A diagonalizable? Why or why not. 84
1. 4 2 0 A-1 1 1 0 0 3 (a) Find the characteristic polynomial of A. (b) What are A's eigenvalues? (c) Find the corresponding eigenvectors (d) Is A diagonalizable? Why or why not. 84
find the eigen space of 4a and 4c
Find the characteristic equations of the following matrices 4. (a) 「 4 0 1 -2 1 0 -2 0 1 (b) [3 0-5 1 1-2 11 1-2 0 (c) 19 5 -4 (d) -1 0 11 -1 3 0 -4 13 1 (e) 5 0 11 ind bases for the eigenspaces of the matrices in Exercise 4 6.
Find the characteristic equations of the following matrices 4. (a) 「 4 0 1...
ex 1.3
Ex. 1.3. For the following 3 x 3 matrices find their characteristic equations, their eigenvalues and the corresponding eigenvectors [2 2 11 4 0 1 5 0 1 5 1 (a)2 1 0; (b) 3 1(d) 1(c) 1 1 2 1 1 0 -7 1 -2 0 1 -2 1 2 2
Ex. 1.3. For the following 3 x 3 matrices find their characteristic equations, their eigenvalues and the corresponding eigenvectors [2 2 11 4 0 1 5...