In what follows we will assume each matrix $A \in \Cmn$ and that $p=\min(m,n)\text{.}$

Recall the Frobenius norm of a matrix:

\begin{equation*} \|A\|_F := \big(\sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2\big)^{1/2}. \end{equation*}

We concluded with noting that $A$ can be written as the sum of rank 1 matrices, and idea we will return to in the next class.

# Subsection22.1Exercises

##### Exercise22.5
• Suppose that $D$ is a diagonal matrix; i.e., $D=\diag(d_{11}, d_{22}, \dots, d_{mm})\text{.}$ Show that each $d_{ii}$ is an eigenvalue of $D\text{.}$ What are the corresponding eigenvectors?
• Suppose that $L$ is a lower triangular matrix. Show that each $l_{ii}$ is an eigenvalue of $L\text{.}$