Section 5.2 Orthogonal Diagonalization

Theorem: The following conditions are equivalent for an $n\times n$ matrix $U$.1. $U$ is invertible and $U^{-1}=U^T$.

2. The rows of $U$ are orthonormal.

3. The columns of $U$ are orthonormal.

 

Proof: If $U$ is an $n\times n$ matrix with orthonormal columns then $U$ has orthonormal rows. Because $U$ is invertible, and $U^T=U^{-1}$ and $U{U}^T=I$.

 

Definition: An orthogonal matrix is a square invertible matrix $U$ such that  $U^{-1}=U^T$.

 

 

Definition: A symmetric matrix is a matrix $A$ such that $A=A^T$.

 

Remark: Such a matrix is necessarily square. Its main diagonal entries are arbitrary, but its other entries occur in pairs — on opposite sides of the main diagonal.

 

Theorem: If $A$ is symmetric, then any two eigenvectors from different eigenspaces are orthogonal.

 

Proof: Use ${\lambda }_1\overrightarrow{v_1}\cdot \overrightarrow{v_2}={\lambda }_2\overrightarrow{v_1}\cdot \overrightarrow{v_2}$.

 

 

Example 1: Find eigenspace of $A=\left[\begin{array}{cc}16& -4\\ -4& 1\end{array}\right]$ and verify the eigenvectors from different eigenspaces are orthogonal.

 

 

Exercise 1: Find eigenspace of $A=\left[\begin{array}{cc}-7& 24\\ 24& 7\end{array}\right]$ and verify the eigenvectors from different eigenspaces are orthogonal.

 

Definition: An $n\times n$ matrix $A$ is said to be orthogonally diagonalizable if there are an orthogonal matrix $P$ (with $P^{-1}=P^T$ and $P$ has orthonormal columns) and a diagonal matrix $D$ such that $A=PD{P}^T=PD{P}^{-1}$.

 

Remark: Such a diagonalization requires $n$ linearly independent and orthonormal eigenvectors. If $A$ is orthogonally diagonalizable, then $A^T=(PD{P}^T{)}^T=(P^T{)}^T{D}^T{P}^T=PD{P}^T=A$,

i.e. $A$ is symmetric.

 

Theorem: An $n\times n$ matrix A is orthogonally diagonalizable if and only if $A$ is symmetric
matrix.

 

Example 2: Orthogonally diagonalize the matrix $A=\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]$.

 

 

Exercise 2: Orthogonally diagonalize the matrix $A=\left[\begin{array}{cc}1& 5\\ 5& 1\end{array}\right]$.

 

Example 3: Orthogonally diagonalize the matrix $A=\left[\begin{array}{ccc}3& -2& 4\\ -2& 6& 2\\ 4& 2& 3\end{array}\right]$.

 

 

Exercise 3: Orthogonally diagonalize the matrix $A=\left[\begin{array}{ccc}5& -4& -2\\ -4& 5& 2\\ -2& 2& 2\end{array}\right]$.

 

Remark: The set of eigenvalues of a matrix $A$ is sometimes called the spectrum of $A$, and the following description of the eigenvalues is called a spectral theorem.

 

Theorem: The Spectral Theorem for Symmetric Matrices

An $n\times n$ symmetric matrix $A$ has the following properties:

(a) $A$ has $n$ real eigenvalues, counting multiplicities.

(b) The dimension of the eigenspace for each eigenvalue $\lambda$ equals the multiplicity of $\lambda$ as a root of the characteristic equation.

(c) The eigenspaces are mutually orthogonal, in the sense that eigenvectors corresponding to different eigenvalues are orthogonal.

(d) $A$ is orthogonally diagonalizable.

 

 

Example 4: Orthogonally diagonalize the matrix $A=\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 2& 0\\ 0& 1& 0& 1\end{array}\right]$.

 

 

Exercise 4: Orthogonally diagonalize the matrix $A=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 3& 0& 0\\ 0& 0& 3& 0\\ 1& 0& 0& 1\end{array}\right].$

 

GroupWorkExample 1: True or False.

 

a. An $n\times n$ matrix that is orthogonally diagonalizable must be symmetric.

 

b. If $A=A^T$ and if vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ satisfy
$A\overrightarrow{u}=3\overrightarrow{u}$ and $A\overrightarrow{v}=4\overrightarrow{v}$ then
$\overrightarrow{u}\cdot \overrightarrow{v}=0$.

 

c. An $n\times n$ symmetric matrix has $n$ distinct real eigenvalues.

 

d. Every symmetric matrix is orthogonally diagonalizable.

 

e. If $B=PD{P}^T$, where $P^T=P^{-1}$ and $D$ is a diagonal matrix, then $B$ is a symmetric matrix.

 

f. The dimension of an eigenspace of a symmetric matrix equals the multiplicity of the corresponding eigenvalue.

 

GroupWork 2: Show that if $A$ and $B$ are orthogonal matrices then $AB$ is also an orthogonal matrix.

 

GroupWork 3: Suppose $A$ is invertible and orthogonal diagonalizable. Show that $A^{-1}$ is also orthogonal diagonalizable.

 

GroupWork 4: Prove the statement or give a counterexample.

 

a. An orthogonal matrix is orthogonally diagonalizable.

 

b. An orthogonal matrix is invertible.

 

c. An invertible matrix is orthogonal.

 

d. If a matrix is diagonalizable then it is symmetric.

 

GroupWork 5: Suppose $A$ is a symmetric $n\times n$ matrix and $B$ is any $n\times m$ matrix. Show that $B^{T}AB$, $B^{T}B$, and $B{B}^T$ are symmetric matrices.