행렬(Matrix) 2. 행렬의 연산

■ 행렬의 덧셈 : 두 행렬 $A$와 $B$가 같은 크기인 경우 그들의 대응 원소를 더함

    $A$와 $B$가 $m \times n$ 행렬이면, $A+B = (a_{ij}) + (b_{ij})_{m \times n}$

        $A = \begin{bmatrix} -4 & 6 & 3 \\ 0 & 1 & 2 \end{bmatrix}$,    $B = \begin{bmatrix} 5 & -1 & 0 \\ 3 & 1 & 0 \end{bmatrix}$ 일때

        $A+B = \begin{bmatrix} -4+5 & 6+(-1) & 3+0 \\ 0+3 & 1+1 & 2+0 \end{bmatrix} = \begin{bmatrix} 1 & 5 & 3 \\ 3 & 3 & 2 \end{bmatrix}$

 

■ 행렬의 스칼라 배(스칼라 곱) : 행렬 $A$에 실수 배(곱)

    $k$가 실수이면, 행렬 $A$의 스칼라 배는

        $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$,    $kA = \begin{bmatrix}k a_{11} &k a_{12} & \cdots &k a_{1n} \\k a_{21} &k a_{22} & \cdots &k a_{2n} \\ \vdots & & & \vdots \\k a_{m1} &k a_{m2} & \cdots &k a_{mn} \end{bmatrix}$

        행렬 $A$의 각 원소에 $k$를 곱한 것 = $(k a_{ij})_{m \times n}$

        $k A = A k$

        두 행렬의 차

        $A-B = A+(-B)$, 여기서 $-B = (-1)B$

 

행렬의 덧셈과 스칼라 곱의 특징

$A$, $B$와 $C$가 $m \times n$ 행렬, $k_{1}$과 $k_{2}$는 스칼라

    (1) $A+B = B+A$    (덧셈의 교환 법칙)

    (2) $A+(B+C) = (A+B)+C$    (덧셈의 교환 법칙)

    (3) $(k_{1} k_{2})A = k_{1} (k_{2} A)$

    (4) $1A = A$

    (5) $k_{1}(A+B) = k_{1}A + k_{2}B$    (분배 법칙)

    (6) $(k_{1} = k_{2} ) A = k_{1} A + k_{2} B$    (분배 법칙)

 

■ 행렬의 곱

$A$가 $m \times p$ 행렬, $B$가 $p \times n$ 행렬이면 $AB$는 $m \times n$ 행렬

        $A_{m \times p}\; B_{p \times n} = C_{m \times n}$

        $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1p} \\ a_{21} & a_{22} & \cdots & a_{2p} \\ \vdots & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mp} \end{bmatrix}$,    $B = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & & \vdots \\ a_{p1} & a_{p2} & \cdots & a_{pn} \end{bmatrix}$

        $C = AB = \begin{bmatrix} a_{11} b_{11} + a_{12} b_{21} + \cdots + a_{1p} b_{p1} & \cdots & a_{11} b_{1n} + a_{12} b_{2n} + \cdots + a_{1p} b_{pn} \\ a_{21} b_{11} + a_{22} b_{21} + \cdots + a_{2p} b_{p1} & \cdots & a_{21} b_{1n} + a_{22} b_{2n} + \cdots + a_{2p} b_{pn} \\ \vdots & & \vdots \\ a_{m1} b_{11} + a_{m2} b_{21} + \cdots + a_{mp} b_{p1} & \cdots & a_{m1} b_{1n} + a_{m2} b_{2n} + \cdots + a_{mp} b_{pn} \end{bmatrix}$

        $C_{ij} = \left [ \sum_{k=1}^{p} a_{ik} b_{kj} \right ]_{m \times n} $, 행에 열을 곱하는 것

        $i = 1, \cdots , m$, $j = 1, \cdots , n$

        $A = \begin{bmatrix} 3 & 5 & -1 \\ 4 & 0 & 2 \\ -6 & -3 & 2\end{bmatrix}$,     $B = \begin{bmatrix} 2 & -2 & 3 & 1 \\ 5 & 0 & 7 & 8 \\ 9 & -4 & 1 & 1 \end{bmatrix}$

        $A_{3 \times 3} \; B_{3 \times 4} = C_{3 \times 4}$

        $C_{3\times4} = \begin{bmatrix} C_{11} & C_{12} & C_{13} & C_{14} \\ C_{21} & C_{22} & C_{23} & C_{24} \\ C_{31} & C_{32} & C_{33} & C_{34}\end{bmatrix} = \begin{bmatrix} 22 & -2 & 43 & 42 \\ 26 & C_{22} & C_{23} & C_{24} \\ -9 & C_{32} & C_{33} & -28\end{bmatrix} $

        $C_{11} = 3 \cdot 2 + 5 \cdot 5 + -1 \cdot 9 = 22$

        $C_{12} = 3 \cdot 2 + 5 \cdot 0 + -1 \cdot -4 = -2$

        $C_{13} = 3 \cdot 3 + 5 \cdot 7 + -1 \cdot 1 = 43$

        $C_{14} = 3 \cdot 1 + 5 \cdot 8 + -1 \cdot 1 = 42$

 

        $C_{21} = 4 \cdot 2 + 0 \cdot 5 + 2 \cdot 9 = 26$

        $C_{31} = -6 \cdot 2 + -3 \cdot 5 + 2 \cdot 9 = -9$

        $C_{34} = -6 \cdot 1 + -3 \cdot 8 + 2 \cdot 1 = -28$

 

○ 행렬 곱셈의 원리(1)

        $y_{1} = a_{11} x_{1} + a_{12} x_{2}$        (1)

        $y_{2} = a_{21} x_{1} + a_{22} x_{2}$

        $y = Ax = \begin{bmatrix} y_{1} \\ y_{2}\end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2}\end{bmatrix} = \begin{bmatrix} a_{11} x_{1} + a_{12} x_{2} \\ a_{21} x_{1} + a_{22} x_{2} \end{bmatrix}$        (2)

        여기서, $x$가 $Bw$ 연관

        $x = Bw = \begin{bmatrix} x_{1} \\ x_{2}\end{bmatrix} = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22}\end{bmatrix} \begin{bmatrix} w_{1} \\ w_{2}\end{bmatrix} = \begin{bmatrix} b_{11} w_{1} + b_{12} w_{2} \\ b_{21} w_{1} + b_{22} w_{2} \end{bmatrix}$

        (2)를 (1)에 대입

        $y = ABw$, $AB$를 $C$행렬로 보면, $y = Cw$  ∴$C = AB$

        $y = Cw = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix} \begin{bmatrix} w_{1} \\ w_{2}\end{bmatrix} = \begin{bmatrix}c_{11} w_{1} + c_{12} w_{2} \\ c_{21} w_{1} + c_{22} w_{2} \end{bmatrix}$        (3)

        $\begin{bmatrix} y_{1} \\ y_{2} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{bmatrix} \begin{bmatrix} b_{11} w_{1} + b_{12} w_{2} \\ b_{21} w_{1} + b_{22} w_{2} \end{bmatrix}$

        $y_{1} = a_{11} (b_{11} w_{1} + b_{12} w_{2} ) + a_{12} (b_{21} w_{1} + b_{22} w_{2} ) = (a_{11} b_{11} + a_{12} b_{21} w_{1} + (a_{11} b_{12} + a_{12} b_{22}) w_{2}$

        $y_{2} = a_{21} (b_{11} w_{1} + b_{12} w_{2} ) + a_{22} (b_{21} w_{1} + b_{22} w_{2} ) = (a_{21} b_{11} + a_{12} b_{21} w_{1} + (a_{21} b_{12} + a_{22} b_{22}) w_{2}$

        식(3)과 비교하면

        $c_{11} = a_{11} b_{11} + a_{12} b_{21}$,        $c_{12} = a_{11} b_{12} + a_{12} b_{22}$

        $c_{21} = a_{21} b_{11} + a_{22} b_{21}$,        $c_{22} = a_{21} b_{12} + a_{22} b_{bb}$

 

○ 행렬 곱셈의 원리(2)

        연립방정식

        $\begin{matrix} ax + by = x' \\ cx + dy = y' \end{matrix}$      $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}$        (1)

        $\begin{matrix} x + 2y = x' \\ 3x + 4y = y' \end{matrix}$  →    $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}$        (2)

        $\begin{matrix} 5x' + 6y' = x'' \\ 7x' + 8y' = y'' \end{matrix}$  →    $\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x'' \\ y'' \end{bmatrix}$        (3)

        $\begin{matrix} 5x' + 6y' = x'' \\ 7x' + 8y' = y'' \end{matrix}$  ←    $\begin{matrix} x + 2y = x' \\ 3x + 4y = y' \end{matrix}$

        $5(x + 2y) + 6(3x + 4y) = (5 \times 1 + 6 \times 3) x + (5 \times 2 + 6 \times 4)y = x''$

        $7(x + 2y) + 8(3x + 4y) = (7 \times 1 + 8 \times 3) x + (7 \times 2 + 68 \times 4)y = y''$

        $\begin{bmatrix} 5 \times 1 + 6 \times 3 & 5 \times 2 +  6 \times 4 \\ 7 \times 1 + 8 \times 3 & 7 \times 2 + 8 \times 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x'' \\ y'' \end{bmatrix}$

        (3)에 (2)를 대입

        $\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x'' \\ y'' \end{bmatrix}$

 

행렬의 곱은 덧셈과 달리 교환적이지 않다. 교환 법칙이 성립하지 않는다.

        $AB \neq BA$

        $A = \begin{bmatrix} 4 & 7 \\ 3 & 5 \end{bmatrix}$,   $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $

        $AB = \begin{bmatrix} 4 \times 9 + 7 \times 6 & 4 \times (-2) + 7 \times  \\ 3 \times 9 + 5 \times 6 & 3 \times (-2) + 5 \times 8 \end{bmatrix} = \begin{bmatrix} 78 & 48 \\ 57 & 34 \end{bmatrix}$

        $BA = \begin{bmatrix} 9 & -2  \\ 6 & 8 \end{bmatrix} \begin{bmatrix} 4 & 7  \\ 3 & 5 \end{bmatrix}= \begin{bmatrix} 9 \times 4 + (-2) \times 3 & 9 \times 7 + (-2) \times 5 \\ 6 \times 4 + 8 \times 3 & 6 \times 7 + 8 \times 5 \end{bmatrix} = \begin{bmatrix} 30 & 53 \\ 48 & 82 \end{bmatrix}$

 

행렬 곱셈의 특징

        $AB \neq BA$, 교환법칙 적용 안됨

        $A_{n \times p}$, $B_{p \times r}$, $C_{r \times n}$ 행렬이면 $A(BC) = (AB)C$,  결합법칙

        $B_{r \times n}$, $C_{r \times n}$, $A_{m \times r}$ 행렬이면 $A(B+C) = AB+AC$,  분배법칙

                                                      $(B+C)A = BA+CA$

 

■ 전치 행렬

        $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$,         $A^{T} = \begin{bmatrix} a_{11} & a_{21} & \cdots & a_{m1} \\ a_{12} & a_{22} & \cdots & a_{m2} \\ \vdots & & & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{mn} \end{bmatrix}$

        예를 들어

        $A = \begin{bmatrix} 3 & 2 & -1 \\ 6 & 5 & 2 \\ 2 & 1 & 4 \end{bmatrix}$ 이면,        $A^{T} = \begin{bmatrix} 3 & 6 & 2 \\ 2 & 5 & 2 \\ -1 & 2 & 4 \end{bmatrix}$

        $B = \begin{bmatrix}5 & 3 \end{bmatrix}$,         $B^{T} = \begin{bmatrix}5 \\ 3 \end{bmatrix}$

 

전치 행렬의 성질

        $A$, $B$ 행렬, $k$ 스칼라

        (1) $(A{T}){T} = A $        전치의 전치

        (2) $(A+B)^{T} = A^{T} + B^{T}$        합의 전치

        (3) $(AB)^{T} = B^{T} A{T}$        곱의 전치

        (4) $(kA){T} = kA^{T}$        스칼라 배의 전치

 

        $(A+B+C)^{T} = A^{T} + B^{T} + C^{T}$

        $(ABC)^{T} = C^{T} B^{T} A^{T}$