c) Let Ae R"n be nonsingular and let -be any natural matrix norm on R be...
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 \(b_{i j}=0\)
, 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
Let R be an ED but not a field, with a norm function N R-Z U0 such that N(ab for all a, bE R. (a) Prove that N(1R) (b) If r E R is a unit, show that N(r) 1. (c) If r E R is nonzero, show that N(r) 0. (d) For any r E R, prove that if N(r) 1, then r is a unit N(a)N(b) e) For any r e R if N() is a prime mumber,...
5. This problem is to help you relate many of the topics we have discussed this semester. Fill in the blanks Let A be an n × n matrix. A is nonsingular if and only if (a) The homogeneous linear system A0 has b) A is row equivalent to (c) The rank of A is (d) Theof A are linearly independent (e) Theof A span (f) The (g) N(A) = of R" Of A form a (i) The map V...
Let A be an ( n x n ) matrix, and let Lambda be an eigenvalue of A. Prove that for any scalar Alpha, Lambda + Alpha is an eigenvalue of A + Alpha x I (identity matrix).
3. Let a >0, and for any A E Rnxn, define Aa aA (a) Prove that for any induced matrix norm, K(Ao) (b) Write the formula for det(Aa) in terms of det(A). estimating well/ill-conditioning of matrices. n(A) . Hint: examine IAall and IAal directly. (c) Based on your result from (a) and (b), comment on whether the determinant is useful for 3. Let a >0, and for any A E Rnxn, define Aa aA (a) Prove that for any induced...
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian matrix. PROVE: If A is positive definite the n every eigenvalue of A is positiv e. (c) Let Abe an n X n Hermitian matrix. PROVE: If every eigenvalue of A is positive. Then A is positive definite. (a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian...
definition of Markov matrix and related theorems are showed below 8.4.2Show that the matrix (8.4.21) is a Markov matrix which is not regular. Is A stable? Definition 8.7 Let A = (aij) A satisfies R(n, n) so that aij-0 for i, j = I, . . . , n. If j-1 243 8.4 Markov matrices that is, the components of each row vector in A sum up to 1, then A is called a Markov or stochastic matrix. If there...
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.)
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.