Find a matrix P such that PTAP orthogonally diagonalizes A. Verify that PTAP gives the proper diagonal form. (Enter each matrix

Question

Mazyrski

2 days ago

14

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

Mathematics

1 answer:

Zina [12K] · Answer 1 · 2026-04-07T13:53:29+03:00

the matrix P that you need is P=(1/ $\sqrt{2}$ ) · [[1 1 0 0],[1 -1 0 0],[0 0 1 1],[0 0 1 -1]]Step-by-step breakdown:1º) Identify the eigenvalues of the matrix in a specific sequence.2º) Determine the eigenvectors corresponding to each eigenvalue.Helpful Note: The matrix A can be expressed as A = $P^{t}$ D P3º) The matrix D is diagonal and contains the eigenvalues arranged in order.4º) The matrix P should consist of the normalized eigenvectors listed in the correct sequence (as columns).Another Note: Keep in mind that $P^{t}$ ·P = I; therefore, if A = $P^{t}$ D P, then:P A $P^{t}$ = P $P^{t}$ D P $P^{t}$ = I D I = DThus, the P we seek is generated from the diagonalization process.Important Advice: Here, A is a block matrix with zero non-diagonal submatrices, which allows us to derive the eigenvalues using the diagonal submatrices alone. This reduces the problem to calculating the eigenvalues of A₁₁ = A ₂₂ = [[5 3],[3 5]]Working through the solution:1º) The eigenvalues of A₁₁ are {8,2}, hence 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}, leading to normalized eigenvectors expressed as P = (1/ $\sqrt{2}$ ) · [[1 1 0 0],[1 -1 0 0],[0 0 1 1],[0 0 1 -1]] (you can observe that P = $P^{t}$ in this instance).In reference to "tip 2": P A $P^{t}$ = P $P^{t}$ D P $P^{t}$ = I D I = DTherefore, the P obtained is indeed what you need.