Math 303: Section 31

Dr. Janssen

\[ \def\R{{\mathbb R}} \def\P{{\mathbb P}} \def\B{{\mathcal B}} \def\C{{\mathcal C}} \def\S{{\mathcal S}} \def\b{{\mathbf{b}}} \def\a{{\mathbf{a}}} \def\c{{\mathbf{c}}} \def\x{{\mathbf{x}}} \def\y{{\mathbf{y}}} \def\u{{\mathbf{u}}} \def\v{{\mathbf{v}}} \def\w{{\mathbf{w}}} \def\z{{\mathbf{z}}} \def\e{{\mathbf{e}}} \def\r{{\mathbf{r}}} \def\M{{\mathcal{M}}} \DeclareMathOperator{\null}{Nul} \DeclareMathOperator{\span}{Span} \DeclareMathOperator{\dim}{dim} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\row}{Row} \DeclareMathOperator{\col}{Col} \DeclareMathOperator{\trace}{trace} \newcommand{\set}[1]{\left\{ {#1} \right\}} \newcommand{\setof}[2]{{\left\{#1\,\colon\,#2\right\}}} \newcommand{\norm}[1]{{\left|\! \left| #1 \right| \! \right|}} \newcommand{\ip}[1]{{\left\langle #1 \right\rangle}} \newcommand{\tr}[1]{{{#1}^\textsf{T}}} \]

An \(n\times n\) matrix \(A\) is **orthogonally diagonalizable** if there is an orthogonal matrix \(P\) such that \(\tr{P} A P\) is a diagonal matrix.

**Observation.** If \(A\) is orthogonally diagonalizable, \(A = \tr{A}\).

Let \(A\) be a symmetric \(n\times n\) matrix and let \(\x,\y \in \R^n\).

- Show that \(\tr{\x} A \y = \tr{(A\x)} \y\).
- Show that \((A\x)\cdot\y = \x\cdot(A\y)\).
- Show that the eigenvalues of a \(2\times 2\) symmetric matrix \(A = \left[\begin{matrix} a & b \\ c & d \end{matrix}\right]\) are real.

**Theorem 1** Let \(A\) be an \(n\times n\) symmetric matrix with real entries. Then the eigenvalues of \(A\) are real.

Let \(A\) be a real symmetric matrix with eigenvalues \(\lambda_1\) and \(\lambda_2\) and corresponding eigenvectors \(\v_1\) and \(\v_2\), respectively.

- Use Activity 31.1 to show that \(\lambda_1 \v_1 \cdot \v_2 = \lambda_2 \v_1 \cdot \v_2\).
- Explain why the result of the previous part shows that \(\v_1\) and \(\v_2\) are orthogonal if \(\lambda_1 \ne \lambda_2\).

**Theorem 2** If \(A\) is a real symmetric matrix, then eigenvectors corresponding to distinct eigenvalues are orthogonal.

Let \(A\) be a real symmetric matrix. Then \(A\) is orthogonally diagonalizable.

**Definition 1** The set of eigenvalues of a matrix A is called the spectrum of A.

**Theorem 3 (The Spectral Theorem for Real Symmetric Matrices)** Let \(A\) be an \(n\times n\) symmetric matrix with real entries. Then:

- \(A\) has \(n\) real eigenvalues (counting multiplicities).
- The dimension of each eigenspace of \(a\) is the multiplicity of the corresponding eigenvalue as a root of the characteristic polynomial.
- Eigenvectors corresponding to different eigenvalues are orthogonal.
- \(A\) is orthogonally diagonalizable.

Let \(A = \left[\begin{matrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{matrix}\right]\). The eigenvalues of \(A\) are 2 and 8, with eigenspaces of dimensions 2 and 1, respectively.

- Explain why \(A\) can be orthogonally diagonalized.
- Two linearly independent eigenvectors for \(A\) corresponding to the eigenvalue 2 are \(\v_1 = \left[\begin{matrix} -1 \\ 0 \\ 1 \end{matrix}\right]\) and \(\v_2 = \left[\begin{matrix} -1 \\ 1 \\ 0 \end{matrix}\right]\). Note that \(\v_1,\v_2\) are not orthogonal, so cannot be in an orthogonal basis of \(\R^3\) consisting of eigenvectors of \(A\). So find a set \(\set{\w_1, \w_2}\) of orthogonal eigenvectors of \(A\) so that \(\span \set{\w_1, \w_2} = \span \set{\v_1, \v_2}\).
- The vector \(\v_3 = \left[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}\right]\) is an eigenvector for \(A\) corresponding to the eigenvalue 8. What can you say about the relationship between the \(\w_i\)’s and \(\v_3\)?
- Find a matrix \(P\) that orthogonally diagonalizes \(A\). Verify your work.

Let \(A\) be an \(n\times n\) symmetric matrix with real entries, and let \(\set{\u_1, \ldots, \u_n}\) be an orthonormal basis of eigenvectors of \(A\) with \(A\u_i = \lambda_i \u_i\) for each \(i\). Further, let \(P_i = \u_i \tr{\u_i}\). Then:

- \(A = \lambda_1 P_1 + \cdots + \lambda_n P_n\)
- \(P_i\) is symmetric
- \(P_i\) is rank 1
- \(P_i^2 = P_i\)
- \(P_i P_j = 0\) if \(i\ne j\)
- \(P_i \u_i = \u_i\)
- \(P_i \u_j = \mathbf{0}\) if \(i\ne j\)
- For any vector \(\v\) in \(\R^n\), \(P_i \v = \proj_{\span\set{\u_i}} \v\).

Let \(A = \left[\begin{matrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{matrix}\right]\). The eigenvalues of \(A\) are \(\lambda_1 = \lambda_2 = 2\) and \(\lambda_3 = 8\), with eigenspaces of dimensions 2 and 1, respectively.

A basis for \(E_8\) is given by \(\set{\tr{[ 1 \ 1 \ 1]}}\) and a basis for \(E_2\) is \(\set{\tr{[1 \ -1 \ 0]}, \tr{[1 \ 0 \ -1]}}\).

- Find orthonormal vectors \(\u_1, \u_2, \u_3\) corresponding to \(\lambda_1, \lambda_2, \lambda_3\), respectively.
- Compute \(\lambda_1 \u_1 \tr{\u_1}\).
- Compute \(\lambda_2 \u_2 \tr{\u_2}\).
- Compute \(\lambda_3 \u_3 \tr{\u_3}\).
- Verify that \(A = \lambda_1 \u_1 \tr{\u_1} + \lambda_2 \u_2 \tr{\u_2} + \lambda_3 \u_3 \tr{\u_3}\).