Find a basis for the eigenspace corresponding to the eigenvalue. 3 1 - 1 A 35-3 66 -4 λ = 2 A basis for the eigenspace corresponding to λ = 2 is. (Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element. Use a comma to separate answers as needed.)

Find the eigenvalues of the symmetric matrix. (Enter your answers as a comma-separated list. Enter your answers from smallest to largest.) 077 707 77 7 λ; = For each eigenvalue, find the dimension of the corresponding eigenspace. (Enter your answers as a comma-separated list.) dim(x;) =

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 11 and -7 18 4 81 4 -5 Enter the matrices P and D below. (Use a comma to separate matrices as needed. Type exact answers, using radicals as needed. Do not label the matrices) 4

For the matrix, list the real eigenvalues, repeated according to their multiplicities. The real eigenvalues are (Use a comma to separate answers as needed.) 8-40 8 6 5 -5 03 8 00 8 0 0 0