Question

Find a matrix P such that PTAP orthogonally diagonalizes A. Verify that PTAP gives the proper diagonal form. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.) A = 5 3 0 0 3 5 0 0 0 0 5 3 0 0 3 5

Answer 1

the P matrix you are looking for is P=(1/
`√(2)`) · [[1 1 0 0],[1 -1 0 0],[0 0 1 1],[0 0 1 -1]]

Explanation:

Answer:

For an orthogonal diagonalization of any matrix you have to:

1º) Find the matrix eigenvalues in a set order.

2º) Find the eigenvectors of each respective eigenvalues.

Tip: You can write the matrix A like A =
`P^(t)` D P

3º) D is the diagonal matrix with each eigenvalue (in order) in the diagonal.

4º) Write P as the normalized eigenvectors in order (in columns).

Tip 2: Remember,
`P^(t)`·P = I, so if A =
`P^(t)` D P, then:

P A
`P^(t)` = P
`P^(t)` D P
`P^(t)` = I D I = D

So the P we are looking for is the
`P^(t)` of the diagonalization.

Tip 3: In this case, A is a block matrix with null nondiagonal submatrixes, therefore its eigenvalues can be calculated by using the diagonal submatrixes. The problem is reduced to calculate the eigenvalues of A₁₁ = A ₂₂ = [[5 3],[3 5]]

Solving:

1º)the eigenvalues of A₁₁ are {8,2}, therefore the D matrix is
`\left[\begin{array}{cccc}8&0&0&0\\0&2&0&0\\0&0&8&0\\0&0&0&2\end{array}\right]`

2º) the eigenvectors of A₁₁ are P₈= {[1 1]T} P₂= {[1 -1]T}, therefore normalizing the eigenvectors you obtain P = (1/
`√(2)`) · [[1 1 0 0],[1 -1 0 0],[0 0 1 1],[0 0 1 -1]] (you can see that P =
`P^(t)` in this case).

As said in "tip 2": P A
`P^(t)` = P
`P^(t)` D P
`P^(t)` = I D I = D

So the P obtained is the one you are looking for.