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7.3:  Orthogonal Diagonalization
\v
Equivalent Questions:
\v

$\bullet$ Given an $n \times n$ matrix, does there exist an orthonormal
basis for $R^n$ consisting of eigenvectors of $A$?
\v

$\bullet$ Given an $n \times n$ matrix, does there exist an orthogonal
matrix $P$ such that  $P^{-1}AP$ = $P^TAP$ is a diagonal matrix?
\v

$\bullet$ Is $A$ symmetric?

Defn:  A matrix is {\bf symmetric} if $A = A^T$.

Defn:  An invertible matrix $P$ is {\bf orthogonal} if $P^{-1} = P^T$
\v

Note $P$ is orthogonal if and only if the columns of $P$ are orthonormal.

Defn:   A matrix $A$ is {\bf orthogonally diagonalizable} if there exists an
orthogonal matrix $P$
such
that $P^{-1}AP = D$ where $D$ is a diagonal matrix.

Thm 7.3.1:  If $A$ is an $n \times n$ matrix, then the following are
equivalent:

\vskip -7pt

\hskip 0.1in a.)  $A$ is orthogonally diagonalizable.
\vskip -7pt
\hskip 0.1in b.)  There exists an orthonormal basis for $R^n$ consisting of
eigenvectors
of $A$.
\vskip -7pt
\hskip 0.1in c.)  $A$ is symmetric.


Thm 7.3.2:  If $A$ is a symmetric matrix, then:
\vskip -7pt
a.)  The eigenvalues of $A$ are all real numbers.
\vskip -7pt
b.)  Eigenvectors from different eigenspaces are orthogonal.



To orthogonally diagonalize a symmetric matrix $A$:
\vskip -5pt

1.) Find the eigenvalues of $A$.

\centerline{Solve $det(A - \lambda I) = 0$ for $\lambda$.}
\vskip -3pt

2.) Find a basis for each of the eigenspaces.

\centerline{Solve $(A - \lambda_j I){\bf x}= 0$ for ${\bf x}$.}
\vskip -3pt

3.) Use the Gram-Schmidt process to find an orthonormal basis for each eigenspace.
\vskip -3pt

4.) Use the eigenvalues of $A$ to construct the diagonal matrix $D$, and use the
orthonormal basis of the corresponding eigenspaces for the corresponding columns
of $P$. \vskip -3pt

5.) Note $P^{-1} = P^T$ since the columns of $P$ are orthonormal.


\end

8.1: 1 - 5 odd, 7, 9, 13, 17, 19, 21, 25, 29, 31, 35, 38, 44