4. Prove that the Vandermonde matrix is nonsingular when I + I; for any i, j...
c) Let Ae R"n be nonsingular and let -be any natural matrix norm on R be an eigenvalue of A. Prove that 1/| A-1|| S AS 11A|l. Let A
Let A be an \(m \times n\) matrix of rank \(r\). Prove that there is a nonsingular \(m \times m\) matrix \(P\) and a nonsingular \(n \times n\) matrix \(Q\) such that the matrix \(B=P A Q=\left(b_{i j}\right)\) has entries \(b_{i i}=1\) for \(1 \leq i \leq r\) and all other entries \(b_{i j}=0\)
5. A is a nonsingular matrix (that is A-exists) and suppose is an eigenvalue of A with associated eigenvector K. 5.1 Prove that 1 70. (Hint: Suppose that i = 0.) 5.2 Show that is an eigenvalue of A-- with corresponding eigenvector K. 5.3 Show that 12 is an eigenvalue of A² with corresponding eigenvector K. (This statement is true even if A is singular.)
Picture enclosed.
(a) Prove that, if A is px p and nonsingular, then the row-echelon form of A I, is 1 A-1 (b) Use this to find the inverse of 2 -3] A =2 4 -1 |-2 1 10 (c) Explain how this result can be used to find the inverse of any nonsingular ma- trix
Hta11 2. Prove that for the (Hilbert) matrix is positive definite. i+j-1 i.j-1 Hnts: (Proceed from the definition to show that if a-(a a in n, then ar Ha>0 .a, is a nonzero vector a 1s a nonzero vector (ii)--= Í xi +j-2 ax (111) manipulate a' Ha into the integral of a positive function. i+ J
Hta11 2. Prove that for the (Hilbert) matrix is positive definite. i+j-1 i.j-1 Hnts: (Proceed from the definition to show that if a-(a...
4. (-12 points) DETAILS LARLINALG8 7.2.009. For the matrix A, find (if possible) a nonsingular matrix P such that p-1AP is diagonal. (If not possible, enter IMPOSSIBLE.) -2 -2 A 0 3-2 0 -1 PE 11 Verify that p-IAP is a diagonal matrix with the eigenvalues on the main diagonal. P-AP Need Help? Read it Talk to a Tutor Submit Answer 5. [-12 Points] DETAILS LARLINALG8 7.2.013. For the matrix A, find (if possible) a nonsingular matrix P such that...
Let A-(Aij)i iJSn є {0,1)"xn denote the symmetric adjacency matrix of an undi- rected graph. For iメj, we have Aij = 1 if entity i and j are connected in a network and 0 otherwise: A 0, i-1,..., n. The stochastic block model (SBM) postulates where is a full rank symmetric K x K connectivity matrix with entries in [0, 1]. a) Consider the matrix P-M MT, where M {0,1)"xK denotes the community k-1,... , K. Show that under (1),...
4 Consider the following nonsingular matrix P = a) Find P by hand. by hand. b) Use P and P-1 to find a matrix B that is similar to A c) Notice that A is a diagonal matrix (a matrix whose entries everywhere besides the main diagonal are 0). As you may recall from #5 on Lab 2, one of the many nice properties of diagonal matrices (of order n) is that 0 1k 0 a11 0 0 a11 0...
a. Prove that if A is any square matrix then A3 X = 23 X (Should prove it for any matrix, this means you cannot choose a specific matrix), where 1 is an eigen value of A and X is its corresponding eigen vector. b. Use the formula in part a to find A X when A = [521
2 0 0 2. Let A be the diagonal matrix 0 4 0First read Exercise 2 of Section 1.5, before continung (a) What would it mean to say that A is nonsingular? (b) Prove that A nonsingular. Give a full explanation using your definition in part Let A be a 4 × 4 matrix with its third row consisting of zeros. (a) What would it mean to say that A is nonsingular? (b) Prove that A is singular. (Hint: Exercise...