Linear Algebra (Math 220)
Assignment due Thursday, May 1

Diagonalization and Orthogonal Diagonalization

Relevant reading: Lay § 7.1

• Two $n\times n$ matrices $A,B$ are called similar when there is an invertible matrix $Q$ such that $Q^{-1}AQ=B$.

• An $n\times n$ matrix $M$ is called diagonalizable when it is similar to a diagonal matrix.

• If $M$ is an $n\times n$ matrix that is the matrix of a linear map $V\stackrel{\phi }{\to }V$ relative to a basis of $V$, then $M$ is diagonalizable if and only if there is some basis of $V$ consisting of eigenvectors of $M$.

• Two $n\times n$ matrices $A,B$ are called orthogonally similar when there is an orthogonal matrix $U$ such that $U^{-1}AU=B$.

• An $n\times n$ matrix $M$ is called orthogonally diagonalizable when it is orthogonally similar to a diagonal matrix.

• If $V$ is a vector space with a given inner product $I$, a linear map $V\stackrel{\phi }{\to }V$ is called symmetric relative to $I$ if and only if for all choices of $v_1,v_2$ in $V$ one has $I\left(\phi \left(v_1\right),v_2\right)=I\left(v_1,\phi \left(v_2\right)\right)$.

• If $M$ is an $n\times n$ matrix that is the matrix of a linear map $V\stackrel{\phi }{\to }V$ with respect to a basis that is orthonormal relative to an inner product $I$, then the following conditions are equivalent:

  1. $M$ is a symmetric matrix.

  2. $\phi$ is symmetric relative to $I$.

  3. $M$ is orthogonally diagonalizable.

  4. There is some orthonormal basis of $V$ consisting of eigenvectors of $\phi$.

Exercises

1. Find a basis of ${\mathbf{R}}^2$ consisting of eigenvectors of the matrix $$\left(\begin{array}{cc}\hfill 5& \hfill 12\\ \hfill 12& \hfill -5\end{array}\right)\text{.}$$

2. Give an example of a $2\times 2$ matrix having eigevalues $1$ and $-1$ where the corresponding eigenvectors form the angle $\pi ⁄4$.

3. Show that the matrix $$\left(\begin{array}{cc}\hfill 2& \hfill 1\\ \hfill 0& \hfill 2\end{array}\right)$$ is not similar to a diagonal matrix.

4. Let $S$ be the $3\times 3$ symmetric matrix $$\left(\begin{array}{ccc}\hfill 2& \hfill -1& \hfill 0\\ \hfill -1& \hfill 3& \hfill -1\\ \hfill 0& \hfill -1& \hfill 2\end{array}\right)\text{.}$$

   1. Find an orthogonal matrix $U$ and a diagonal matrix $D$ such that $$U^{-1}SU=D\text{.}$$

   2. What is the largest value achieved on the unit sphere ${x_1}^2+{x_2}^2+{x_3}^2=1$ by the function $$h\left(x\right)={}^{t}xSx=2{x_1}^2+3{x_2}^2+2{x_3}^2-2x_1{x}_2-2x_2{x}_3\text{?}$$

5. What geometric property might be said to characterize the $n\times n$ matrices that are similar to upper triangular matrices?