Matrix Type

• Identity Matrix
• Upper Triangular Matrix
• Lower Triangular Matrix
• Symmetric Matrix
• Diagonal Matrix
• [[#Skew Symmetric Matrix|Skew Symmetric Matrix]]
• Square Matrix

Permutation Matrix §

A permutation matrix has rows of the identity in any order.

Every row and every column of a permutation matrix contain exactly one nonzero entry, which is 1. There are two 2 × 2 permutation matrices:

$$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\&\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$

Every permutation matrix is a product of elementary row-interchange matrices.

$$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}{x}_2\\ x_1\end{array}\right]$$

Identity Matrix §

$$\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 1& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 1\end{array}\right]$$

Upper Triangular Matrix §

$$\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1n}\\ 0& a_{22}& a_{23}& \dots & a_{2n}\\ 0& 0& a_{33}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & a_{nn}\end{array}\right]$$

Lower Triangular Matrix §

$$\left[\begin{array}{ccccc}{a}_{11}& 0& 0& \dots & 0\\ a_{21}& a_{22}& 0& \dots & 0\\ a_{31}& a_{32}& a_{33}& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& a_{n3}& \dots & a_{nn}\end{array}\right]$$

Symmetric Matrix §

$A=A^T$

$$\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{12}& a_{22}& a_{23}\\ a_{13}& a_{23}& a_{33}\end{array}\right]$$

Diagonal Matrix §

$$\left[\begin{array}{cccc}{d}_1& 0& \dots & 0\\ 0& d_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & d_n\end{array}\right]$$

Skew Symmetric Matrix §

$$\left[\begin{array}{ccc}0& -a_{12}& a_{13}\\ a_{12}& 0& -a_{23}\\ -a_{13}& a_{23}& 0\end{array}\right]$$

Square Matrix §

Number of row = Number of column

$$\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \dots & a_{nn}\end{array}\right]$$