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Linear Algebra with SageMath: A Gentle Introduction

Ajit Kumar

Section 6.4 Orthogonal Diagonalizations

Definition 6.4.2.

A square matrix \(P\) is called an orthogonal matrix if it satisfies any one (and hence all) the conditions of Theorem Theorem 6.4.1.

Example 6.4.3.

(i) The matrix \(\begin{pmatrix}\cos \theta \amp -\sin\theta\\\sin\theta \amp \cos\theta \end{pmatrix}\) is an orthogonal matrix.
(ii) \(\left(\begin{array}{rrr} -\frac{1}{3} \, \sqrt{3} \amp \sqrt{\frac{2}{3}} \amp 0 \\ \frac{1}{3} \, \sqrt{3} \amp \frac{1}{2} \, \sqrt{\frac{2}{3}} \amp -\sqrt{\frac{1}{2}} \\ \frac{1}{3} \, \sqrt{3} \amp \frac{1}{2} \, \sqrt{\frac{2}{3}} \amp \sqrt{\frac{1}{2}} \end{array} \right)\) is an orthogonal matrix.

Definition 6.4.4.

An \(n\times n\) matrix is called orthogonally diagonalizable if there exists an orthogonal matrix \(P\) such that \(P^{-1}AP\) is a diagonal matrix.

Example 6.4.5.

Let \(A\) be a symmetric matrix and \(\lambda_1\) and \(\lambda_2\) are distinct eigenvalues of \(A\text{.}\) If \(v_1\) and \(v_2\) are eigenvectors corresponding to \(\lambda_1\) and \(\lambda_2\) respectively. Then \(v_1\) and \(v_2\) are orthogonal.

Example 6.4.7.

Consider a matrix \(A=\left(\begin{array}{rrr} 5 \amp -2 \amp -4 \\ -2 \amp 8 \amp -2 \\ -4 \amp -2 \amp 5 \end{array} \right)\text{.}\) Clearly \(A\) is symmetric and hence it is orthogonally diagonalizable. The characteristic polynomial of \(A\) is
\begin{equation*} det(xI-A)=x^3 - 18*x^2 + 81*x=x(x-9)^2\text{.} \end{equation*}
Hence \(0, 9, 9\) are eigenvalues of \(A\text{.}\) Its is easy to find that \(v_1=(1, 1/2, 1)\) is an eigenvector corresponding to the eigenvalue 0. \(v_2=(1, 0, -1), v_2=(0, 1, -1/2)\) are eigenvectors corresponding to eigenvalue 9. Hence \(P:=\left(\begin{array}{rrr} 1 \amp 1 \amp 0 \\ \frac{1}{2} \amp 0 \amp 1 \\ 1 \amp -1 \amp -\frac{1}{2} \end{array} \right)\text{.}\) Then
\begin{equation*} P^{-1}AP=\left(\begin{array}{rrr} 0 \amp 0 \amp 0 \\ 0 \amp 9 \amp 0 \\ 0 \amp 0 \amp 9 \end{array} \right)\text{.} \end{equation*}

Problem 6.4.8.

For the following matrices find an orthogonal matrix \(P\) such that \(P^{-1}AP\) is a diagonal matrix.
\begin{equation*} \begin{pmatrix}2 \amp -1 \\-1 \amp 1 \end{pmatrix} , \begin{pmatrix}1 \amp 0 \amp -1\\0 \amp 1 \amp 2\\-1 \amp 2 \amp 5 \end{pmatrix} \end{equation*}
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