Suppose A is a symmetric matrix. (Au) = = ○ AUT Au OUTA ○ ATUT Suppose μ are distinct eigenvalues of A, with corresponding eigenvectors u and v. (Au) Tv = ○ Au Tv Ο μυν ☐ Au Tv (Check ALL that apply) Т u Tv , T Prove your result. Suggestion: If v are column vectors corresponding to the vectors u, v, then uv is a matrix whose entry is u · v. Explain why a symmetric matrix with real entries can't have complex eigenvalues. Suggestion: If A is a matrix with real entries and v is an eigenvector, then must also be an eigenvector. What is vv?

can you please provide explanations

Consider the Matrix

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 10 and -9. A = 1 -3 9 -3 93 9 3 1 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)