Types of Matrix

$$A=[a_{ij}]$$

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

Square Matrix

$$m=n$$ $$\left[\begin{array}{ccc}0& 1& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$$

Rectangular Matrix

$$m\ne n$$ $$\left[\begin{array}{ccc}0& 1& 2\\ -1& 0& 3\\ -2& -3& 0\\ 9& 3& 6\end{array}\right]$$

Zero Matrix

$$all[a_{ij}]=0$$

Symmetric Matrix

$$a_{ij}=a_{ji}$$

Upper-Triangle Matrix

$$a_{ij}=0i>j$$

Lower-Triangle Matrix

$$a_{ij}=0j<i$$ $$\left[\begin{array}{ccc}4& 0& 0\\ 1& 2& 0\\ 6& 8& 3\end{array}\right]$$

Diagonal Matrix

$$a_{ij}=0ifj\ne i$$ $$\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$$

Identity Matrix

$$\begin{gathered} a_{ij}=0ifi\ne j \\ {a}_{ij}=1ifi=j \end{gathered}$$

Skew-Symmetric Matrix

$$a_{ij}=-a_{ji}$$ $$\left[\begin{array}{ccc}0& 1& 2\\ -1& 0& 3\\ -2& -3& 0\end{array}\right]$$

Matrix Transformation

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

$$\begin{gathered} BeforeTransformation \\ \left[\begin{array}{ccc}3& 1& 1\\ -1& 3& 1\\ -1& -1& 3\end{array}\right] \end{gathered}$$ $$\begin{gathered} AfterTransformation \\ \left[\begin{array}{ccc}3& -1& -1\\ 1& 3& -1\\ 1& 1& 3\end{array}\right] \end{gathered}$$

성질

$$2.(A+B{)}^T=A^T+B^T$$ $$3.(cA{)}^T=c{A}^T$$ $$4.(AB{)}^T=B^T{A}^T$$ $$5.if{A}^T=A,AisSymmetric$$ $$6.A^T=-A,A,AisSkew-Symmetric$$ $$7.I^T=I,O^T=O$$