The matrix A is a real symmetric 5 x 5 matrix with eigenvalues 3,7, 10, 15, and 21. The vector æ1 is an eigenvector of A with eigenvalue 3, the vector x, is an eigenvector of A with eigenvalue 7, the vector a3 is an eigenvector of A with eigenvalue 10, the vector 24 is an eigenvector of A with eigenvalue 15, and the vector æ; is an eigenvector of A with eigenvalue 21. Compute the dot products of the eigenvectors a with each other. x1. x2 = x1• 23 = x1. X4 = x2· X3 = x2 · X4 = X2 · X5 = X3 · X4 = x3 · X5 = X4 . X5 =

Let A be a 4 × 4 real matrix. If two of its eigenvalues are A1 = 1- i and A2 then what are the remaining eigenvalues A3 and \4? 4i, Select one: O a. A3 = 1 + i and 4 = -4i O b. A3 = 1 - i and 4 = -4i O c. A3 = 1 + i and 4 = 4i O d. A3 = 1 - i and 4 = 4i

Suppose A has eigenvalue 2 corresponding to eigenvector , and eigenvalue – 2 corresponding to eigenvector The matrix is A =

The matrix A = 4 eigenspace. eigenvalue dimension of the eigenspace (GM) = has one eigenvalue of algebraic multiplicity 2. Find this eigenvalue and the dimenstion of the

The matrix 0 2 -3 -5 -2 -1 -7 has one real eigenvalue. Find this eigenvalue, its multiplicity, and the dimension of the corresponding eigenspace. The eigenvalue = Is the matrix A defective? choose has multiplicity = 6 -5 A = and the dimension of the corresponding eigenspace is

Find a basis of the eigenspace associated with the eigenvalue -3 of the matrix A Answer: -2 -1 1 0 -3 0 0 -5 72 -2 -6 9 6 -6

Give an example of a 3 x 3 matrix with eigenvalues of 2, 2, and -3 that is NOT diagonalizable. Show WHY it is not diagonalizable.

Show that if the entries of each column of an n × n matrix add up to 1, then λ = 1 must bean eigenvalue for the matrix

Find the eigenvalues and a basis for for the eigenspace of the matrix associated with each -2 2 eigenvalue for B = 0 0 0

For a rectangular matrix A, how are the eigenvalues of AT A and AA¹ related?