*Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
5. Let A be an n × n invertible matrix. Show that the condition number of A with respect to the Frobenius norm 5. Let A be an n × n invertible matrix. Show that the condition number of A with respect to the Frobenius norm
Let A be an m x n matrix of unspecified rank. Let b e Rm, and let Prove that this infimum is attained. In other words, prove the existence of an ax for which l|Ax bll p. In this problem, the norm is an arbitrary one defined on Rm. Let A be an m x n matrix of unspecified rank. Let b e Rm, and let Prove that this infimum is attained. In other words, prove the existence of an...
Prove that trace (A*A)=||A||F2 Definition. Let A be an m x n complex matrix. Its Frobenius noTm, denoted by ||A||F, is defined as follows: m ΣΣΑ Β) / . A|| F = j=1 k=1 1. (7 marks) Let A be as above. Prove that trace(AH A) || A||?
8. Let A be an nxn matrix with distinct n eigenvalues X1, 2... (a) What is the determinant of A. (b) If a 2 x 2 matrix satisfies tr(AP) = 5, tr(A) = 3, then find det(A). (The trace of a square matrix A, denoted by tr(A), is the sum of the elements on the main diagonal of A.
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...
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14] QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
, then n lim Let Ά be a square matrix. Prove that if ρ(A)<1 Use the following fact without proof. For any square matrix A and any positive real number ε , there exists a natural matrix norm I l such that l-4 ll < ρ (d) +ε IIA" 11-0
b. Let B be the matrix defined as co 1+cAtcA..A. Prove that the eigenvalues of B are given by (5 pe
2. Let A be an n x n matrix with AT =-A (a) Prove that A has value 0. (b) Prove that A has determinant 0 if n is odd.