Find the eigenvalues A₁ < A2 and associated unit eigenvectors 1, 2 of the symmetric matrix The smaller eigenvalue A₁ = 3 14 A = 14 -18 has associated unit eigenvector i == The larger eigenvalue №2 = has associated unit eigenvector 2 = Note: The eigenvectors above form an orthonormal eigenbasis for A.

A diagonalization of the matrix A is given in the form P-¹AP = D. List the eigenvalues of A and bases for the corresponding eigenspaces. (Repeated eigenvalues should be entered repeatedly with the same eigenspaces.) d₁= = ^₂ = 13 = 1007240 18 1007/N +10 18 18 2 1 4 1 2 5 9 4 4 4 0 0 1 4 4 3 9 1 0 1 -1 1 8 0 -1 has eigenspace span has eigenspace span has eigenspace = 80 0 00 0 0 0 -1 (LF) span (smallest λ-value) (largest λ-value)

Find the eigenvalues

The matrix 4 0 0 A 0 4-8 -4 4 -8] has eigenvalues -4, 0, and 4. Find its eigenvectors. The eigenvalue -4 has associated eigenvector The eigenvalue 0 has associated eigenvector The eigenvalue 4 has associated eigenvector 9.

Find the eigenvalues A1 < A2 < A3 and associated unit eigenvectors ủ1, ü2, ūz of the symmetric matrix 4 -2 -2] A = -2 -2 4 -2 4 -2 The eigenvalue X1 = 2 has associated unit eigenvector ū1 The eigenvalue A2 = has associated unit eigenvector ü2 =

Find the eigenvalues and eigenvectors of A = Show that the eigenvectors are orthog- onal.

Find the eigenvalues of the matrix Smaller eigenvalue = Associated eigenvector = Larger eigenvalue = Associated eigenvector = -12 -8 40 24 Note: vectors are entered with "angle brackets", such as or .