Suppose A is a real symmetric matrix whose eigenvalues are A₁ = 1, A2 3, and X3 ei = Find A. A = = 9 e₂ = - 0 1 √2 1 √2 " = and es - 1, with corresponding eigenvectors = 1 √2 1

Find the matrix A of the quadratic form associated with the equation. 3x2 - 8xy - 3y² + 15 = 0 A Find the eigenvalues of A. (Enter your answers as a comma-separated list.) Find an orthogonal matrix P such that P'AP is diagonal. (Enter the matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.) P =

10 k Q11. Show that the matrix 0 1 is similar to an 0 0 1 orthogonal matrix if and only if k = 0. use both spatial/geometric intuition and eigenvalues/eigenspaces.

If matrix A is square of size n x n with a rank equal to 2 and n > 2. The rank of a matrix is equal to the number of non-zero eigenvalues. Can you write A as the product of two matrices B E Rnxq and C E RPX" where we require that p < n and q < n? You may assume that A is diagonalizable. Show how to construct a matrix B and C such that A = BC.