Types of Matrix

A;=;\[a\_{ij}\]

• $i=m$ , row size
• $j=n$, column size

Square Matrix

$$m=n$$ \\begin{bmatrix} 0 & 1 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{bmatrix}

Rectangular Matrix

$$m;\ne ;n$$ \\begin{bmatrix} 0 & 1 & 2\\ -1 & 0 & 3\\ -2 & -3 & 0\\ 9 & 3 & 6 \end{bmatrix}

Zero Matrix

all;\[a\_{ij}\];=;0

Symmetric Matrix

$$a\_ij;=;a\_ji$$

Upper-Triangle Matrix

$$a\_ij;=;0;;;;i>j$$

Lower-Triangle Matrix

$$a\_ij=0j\text{<}i$$ \\begin{bmatrix} 4 & 0 & 0\\ 1 & 2 & 0\\ 6 & 8 & 3 \end{bmatrix}

Diagonal Matrix

$$a\_ij;=;0;;;;if;j\ne i$$ \\begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{bmatrix}

Identity Matrix

$$\begin{gathered} a\_ij;=;0;;;;if;;i\ne j \\ a\_ij;=;1;;;;if;;i=j \end{gathered}$$

Skew-Symmetric Matrix

$$a\_ij;=;-a\_ji$$ \\begin{bmatrix} 0 & 1 & 2\\ -1 & 0 & 3\\ -2 & -3 & 0 \end{bmatrix}

Matrix Transformation

전치 행렬은 정사각행렬에서, 행과 열의 숫자가 같은 요소들을 기준으로 하는 선으로 행렬을 대칭 시킨 것을 말한다.

$$\begin{gathered} Before;Transformation \\ \left[\begin{array}{ccc}3& 1& 1\\ -1& 3& 1\\ -1& -1& 3\end{array}\right] \end{gathered}$$ $$\begin{gathered} After;Transformation \\ \left[\begin{array}{ccc}3& -1& -1\\ 1& 3& -1\\ 1& 1& 3\end{array}\right] \end{gathered}$$

성질

$$\begin{gathered} 2.;(A+B{)}^T;=;A^T+B^T \\ ; \end{gathered}$$ $$\begin{gathered} 3.;(cA{)}^T;=;c{A}^T \\ ; \end{gathered}$$ $$\begin{gathered} 4.;(AB{)}^T;=;B^T{A}^T \\ ; \end{gathered}$$ $$\begin{gathered} 5.;if;;A^T;=;A;,;A;is;Symmetric \\ ; \end{gathered}$$ $$\begin{gathered} 6.;A^T;=;-A;,;A;,;A;is;Skew-Symmetric \\ ; \end{gathered}$$ $$7.;I^T;=;I;,;O^T;=;O$$