Normal Matrices

Is $\mathbf{A} = \begin{pmatrix} 5+i & -2i \\ 2 & 4+2i \end{pmatrix}$ a normal matrix?

Give examples of two distinct classes of normal matrices that are real but not symmetric.

Show that $\mathbf{A} \in \mathbb {R}^{n \times n}$ is normal and has real eigenvalues if and only if $\mathbf{A}$ is symmetric.

Prove that the eigenvalues of a real skew-symmetric or skew-hermitian matrix must be pure imaginary numbers (i.e., multiples of $i$).

When trying to decide what's true about matrices and what's not, it helps to think in terms of the following associations.

For example, the complex function $f(z) = (1-z)(1+z)^{-1}$ maps the imaginary axis in the complex plane to points on the unit circle because $\left|f(z)\right|^2=1$ whenever $\overline{z}=-z$. It's therefore reasonable to conjecture (as Cayley did in 1846) that if $\mathbf{A}$ is skew hermitian (or real skew symmetric), then

is unitary (or orthogonal). Prove this is indeed correct.

Show by example that a normal matrix can have a complete independent set of eigenvectors that are not orthonormal, and then explain how every complete independent set of eigenvectors for a normal matrix can be transformed into a complete orthonormal set of eigenvectors.

Construct an example to show that the converse of the statement ``if $\mathbf{A}$ is normal, then $N(\mathbf{A}-\lambda_i\mathbf{I}) \perp N(\mathbf{A}-\lambda_j\mathbf{I})$ whenever $\lambda_i \ne \lambda_j$'' is false. In other words, show that it is possible for $N(\mathbf{A}-\lambda_i\mathbf{I}) \perp N(\mathbf{A}-\lambda_j\mathbf{I})$ whenever $i \ne j$ without $\mathbf{A}$ being normal.

Explain why a triangular matrix is normal if and only if it is diagonal.

Use the result of Exercise 7.5.8 to give an alternate proof of the unitary diagonalization theorem, which states that $\mathbf{A} \in \mathbb {C}^{n \times n}$ is unitarily similar to a diagonal matrix if and only if $\mathbf{A}^{*}\mathbf{A} = \mathbf{A}\mathbf{A}^{*}$.

For a normal matrix $\mathbf{A}$, explain why $(\lambda , \overrightarrow {x})$ is an eigenpair for $\mathbf{A}$ if and only if $(\overline{\lambda }, \overrightarrow {x})$ is an eigenpair for $\mathbf{A}^{*}$.

To see if you understand the proof of the min-max part of the Courant--Fischer theorem, construct an analogous proof for the max-min part.

The Courant--Fischer theorem, min-max part: if $\mathbf{A} \in \mathbb {C}^{n \times n}$ is hermitian with eigenvalues $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$, then

and the max-min part asserts the analogous formula

The Courant--Fischer theorem has the following alternate formulation, for $1 < i < n$:

To see if you really understand the proof of the min-max part of the Courant--Fischer theorem, adapt it to prove the alternate min-max formulation given above.