Problem 1. Consider the following matrix A over the field C: 1 10-1 0 0 -1...
Bycalculatingthecharacteristicpolynomial,eigenvaluesanddimensionsoftheeigenspaces of each map or matrix below, determine if the given map or matrix is diagonalizable. If a map or matrix is diagonalizable, diagonalize it (linear algebra) 1. By calculating the characteristic polynomial, eigenvalues and dimensions of the eigenspaces of each map or matrix below, determine if the given map or matrix is diagonalizable. If a map or matrix is diagonalizable, diagonalize it (that is, give a basis consisting of its eigenvectors) The field F over which you consider the...
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.
1 Compute and completely factor the characteristic polynomial of the following matrix: 0 A= -4 5 0 1 1 For credit, you have to factor the polynomial and show work for each step. B In the following, use complex numbers if necessary. For each of the following matrices: • compute the characteristic polynomial; • list all the eigenvalues (possibly complex) with their algebraic multiplicity; • for each eigenvalue, find a basis (possibly complex) of the corresponding eigenspace, and write the...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).
2. Consider the matrix 11 2 4 0 0 -1 1 7 0 0 0 6 10 007) Is this matrix diagonalizable? Explain why or why not. 3. Consider the matrix /1 a b 5 0 1 C 3 A = 0 0 1 2 0 0 0 2 For which values of a, b, c E R is A diagonalizable? Justify your answer.
1. Let T be the matrix T=10 3 acting on the complex vector space V C3 (a) Recall how T defines the structure of a C-module on C3. (b) Let p(x71, and let 2Compute the element p(x) v of C3 (c) Give a set of generators and relations for C3 over Cz] with the above module structure. (d) Write down the relations matrix (e) Recall the definition of minimal polynomial of a matrix. (f) What is the minimal polynomial of...
Consider the following matrix 2 0 OY A= 1 2 10 24/ a Does A has an inverse? Why or why not? b. Is A diagonalizable? c. IfA is diagonalizable, find the matrix P that diagonalizes A. d. For your P, what is the diagonal matrix D? (DO NOT find P-1.just write down D) Write down the fundamental solution matrix (t) for the system of ODEs. /2 0 0 1 2X 0 24/ OV X'=
3. Let Z= (3 a 2 x 2 matrix over Zs. Find the characteristic polynomial of Z and determine for which values of h e Z5, Z is diagonalizable.
Q3. Given the following matrices, A=[ 3), B =[10], C= [31 a. Find the characteristic polynomial of A, B, C respectively. b. Is A diagonalizable? Is B diagonalizable? Is C diagonalizable? If no, please state your reason. If yes, please find the matrix P and D such that p-1MP = D c. Is the matrix A similar to the matrix C? Please explain your answer briefly.
Question 3 Consider the matrix A. (rowt:10, 2, -1];row2 2,3,-2): row:(-1, -2,01). 1. Show that V2-vector column [-2. 1. Oj is an eigenvector for A and find its corresponding eigenvalue L1. 2. Diagonalizable A given that its characteristic polynomial is P(L) = -_LA3) + 3"(L^2)+ 9*L+5.