Diagonal and trace
1. Description
1.1 Diagonal
$$\boldsymbol{v}_{i}=A_{i,i},\: \: i=\left \{ 1,2,\cdots ,min(m,n) \right \}$$
1.2 Trace
The trace of an n × n square matrix A is defined as
$$tr(\boldsymbol{A})=\sum_{i=1}^{m}A_{i,i}=\sum_{i=1}^{n}a_{ii}=a_{11}+a_{11}+\cdots +a_{nn}$$
where aii denotes the entry on the ith row as well as ith column of A.
2. Example
2.1 Diagonal
$$diag\left ( \begin{pmatrix}
1 & -1 & 8\\
-1 & -2 & 4\\
0 & 3 & 5
\end{pmatrix} \right )=\begin{pmatrix}
1\\
-2\\
5
\end{pmatrix}$$
$$diag\left ( \begin{pmatrix}
1 & -1\\
-1 & -2\\
0 & 3
\end{pmatrix} \right )=\begin{pmatrix}
1\\
-2
\end{pmatrix}$$
2.2 Trace
$$trace\left ( \begin{pmatrix}
1 & -1 & 8\\
-1 & -2 & 4\\
0 & 3 & 5
\end{pmatrix} \right )=1+(-2)+5=4$$
3. References
https://en.wikipedia.org/wiki/Trace_(linear_algebra)
https://en.wikipedia.org/wiki/Diagonal_matrix