2) Let A be an nxn matrix with eigenvalue a of multiplicity n. Show that A...
2) Let A be an nxn matrix with eigenvalue of multiplicity n. Show that Ais diagonalizable if and only if A = al.
is diagonalizable if 2. Let A be an nxn matrix and B be an m xm matrix. Show that and only if both A and B are diagonalizable.
For the given Matrix B, find: 1. The algebraic multiplicity of each eigenvalue. 2. The geometric multiplicity of each eigenvalue. 3. The matrix B is it Diagonalizable? If YES, provide the matrices P and D. ( 22-1 B = 1 3 -1 (-1 -2 2
is an eigenvalue invertible matrix with X as an eigenvalue. Show that of A-1. Suppose v ER is a nonzero column vector. Let A (a) Show that v is an eigenvector of A correspond zero column vector. Let A be the n xn matrix vvT. n eigenvector of A corresponding to eigenvalue = |v||2. lat O is an eigenvalue of multiplicity n - 1. (Hint: What is rank A?) (b) Show that 0 is an eigenvalue of
A scalar matrix is simply a matrix of the form XI, where I is the nxn identity matrix. (a) Prove that if A is similar 1 to \I, then in fact A= \I. (b) Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix. 1 100 100 (c) Prove that o 100 is not diagonalizable. 0 0 1 1
True False a) For nxn A, A and AT can have different eigenvalues. b) The vector v 0 cannot be an eigenvector of A. c) If λ's an eigenvalue of A, then λ2 is an eigenvalue of A2. True False d) If A is invertible, then A is diagonalizable. e) If nxn A is singular, then Null(A) is an eigenspace of A. f) For nxn A, the product of the eigenvalues is the trace of A. True False g) If...
1) Let A and B be nxn matrices. Show that if I is a nonzero eigenvalue of AB, then it is also an eigenvalue of BA.
ui l uentical . i Let A be a square matrix of order n and λ be an eigenvalue of A with geometric multiplicity k, where 1kn. Choose a basis B -(V1, v2,. .. , Vk) of &A) and extend this to a basis B of R". (1) Show that the matrix of the linear transformation x Ax on R" induced by the matrix A with respect the basis B on both the domain and codomain is: ui l uentical...
estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.
Let A be a nxn matrix and let B be the matrix composed of co-factors of the matrix A. Show that if rk(A) = n - 1 then rk(B) = 1. Explain.