2014/04/18 | Improve Society

1. Equality of Matrices

Two matrices $A = (a_{jk})$ and $B = (b_{jk})$ of the same order [i.e. equal numbers of rows and columns] are equal if and only if $a_{jk} = b_{jk}$ .

2. Addition of Matrices

If $A = (a_{jk})$ and $B = (b_{jk})$ have the same order we define the sum of $A$ and $B$ as $A + B = (a_{jk} + b_{jk})$ .

Note that the communicative and associative laws for addition are satisfied by matrices, i.e. for any matrices $A,\ B,\ C$ of the same order
$A + B = B + A,\ A + (B + C) = (A + B) + C \cdots (2)$
3. Subtraction of Matrices

If $A = (a_{jk})$ , $B = (b_{jk})$ have the same order, we define the difference of $A$ and $B$ as $A - B = (a_{jk} - b_{jk})$ .

4. Multiplication of a Matrix by a Number

If $A = (a_{jk})$ and $\lambda$ is any number or scalar, we define the product of $A$ by $\lambda$ as $\lambda A = A\lambda = (\lambda a_{jk})$ .

5. Multiplication of Matrices

If $A = (a_{jk})$ is an $m\times n$ matrix while $B = (b_{jk})$ is an $n\times p$ matrix, then we define the product $A\cdot B$ or $AB$ as the matrix $C = (c_{jk})$ where
$\displaystyle c_{jk} = \sum_{l = 1}^n a_{jl}b_{lk} \cdots (3)$
and where $C$ is of order $m\times p$ .

Note that in general $AB \ne BA$ , i.e. the communicative law for multiplication of matrices is not satisfied in general. However, the associative and distributive laws are satisfied, i.e.
$A(BC) = (AB)C,\ A(B + C) = AB + AC,\ (B + C)A = BA + CA \cdots (4)$
A matrix $A$ can be multiplied by itself if and only if it is a square matrix. The product $A\cdot A$ can in such case be written $A^2$ . Similarly we define powers of a square matrix, i.e. $A^3 = A\cdot A^2,\ A^4 = A\cdot A^3$ , etc.

6. Transpose of a Matrix

If we interchange rows and columns of a matrix $A$ , the resulting matrix is called the transpose of $A$ and is denoted by $A^T$ . In symbols, if $A = (a_{jk})$ then $A^T = (a_{kj})$ .

We can prove that
$(A + B)^T = A^T + B^T,\ (AB)^T = B^TA^T,\ (A^T)^T = A \cdots(5)$
7. Symmetric and Skew-Symmetric matrices

A square matrix $A$ is called symmetric if $A^T = A$ and skew-symmetric if $A^T = - A$ .

Any real square matrix [i.e. one having only real elements] can always be expressed as the sum of a real symmetric matrix and a real skew-symmetric matrix.

8. Complex Conjugate of a Matrix

If all elements $a_{jk}$ of a matrix $A$ are replaced by their complex conjugates $\bar{a}_{jk}$ , the matrix obtained is called the complex conjugate of $A$ and is denoted by $\bar{A}$ .

9. Hermitian and Skew-Hermitian Matrices

A square matrix $A$ which is the same as the complex conjugate of its transpose, i.e. if $A = \bar{A}^T$ , is called Hermitian. If $A = -\bar{A}^T$ , then $A$ is called skew-Hermitian. If $A$ is real these reduce to symmetric and skew-symmetric matrices respectively.

10. Principal Diagonal and Trace of a Matrix

If $A = (a_{jk})$ is a square matrix, then the diagonal which contains all elements $a_{jk}$ for which $j = k$ is called the principal or main diagonal and the sum of all elements is called trace of $A$ .

A matrix for which $a_{jk} = 0$ when $j \ne k$ is called diagonal matrix.

11. Unit Matrix

A square matrix in which all elements of the principal diagonal are equal to 1 while all other elements are zero is called the unit matrix and is denoted by $I$ . An important property of $I$ is that
$AI = IA = A,\ I^n = I,\ n = 1,2,3,\cdots(6)$
The unit matrix plays a role in matrix algebra similar to that played by the number one in ordinary algebra.

12. Zero or Null matrix

A matrix whose elements are all equal to zero is called the null or zero matrix and is often denoted by $O$ or symply 0. For any matrix $A$ having the same order as 0 we have
$A + 0 = 0 + A = A \cdots(7)$
Also if $A$ and 0 are square matrices, then
$A0 = 0A = 0 \cdots(8)$
The zero matrix plays a role in matrix algebra similar to that played by the number zero of ordinary algebra.

1. 行列が等しい

　二つの行列 $A = (a_{jk})$ および $B = (b_{jk})$ が同じ次数で（すなわち行と列の数が同じで） $a_{jk} = b_{jk}$ の時にのみ 等しい．

2. 行列の和

　仮に $A = (a_{jk})$ および $B = (b_{jk})$ が同じ次数ならば $A$ および $B$ の和を $A + B = (a_{jk} + b_{jk})$ と定義できます．

　行列の交換法則と結合法則は，すなわちある同じ次数の行列 $A,\ B,\ C$ を下記のように記述します．

$A + B = B + A,\ A + (B + C) = (A + B) + C \cdots (2)$

3. 行列の差

　仮に $A = (a_{jk})$ , $B = (b_{jk})$ が同じ次数を有するなら $A$ および $B$ の差を $A - B = (a_{jk} - b_{jk})$ と定義できます．

4. 行列のスカラー倍

　仮に $A = (a_{jk})$ があって $\lambda$ が任意の数またはスカラーの時 $A$ の $\lambda$ による積を $\lambda A = A\lambda = (\lambda a_{jk})$ と定義できます．

5. 行列の積

　仮に $A = (a_{jk})$ が $m\times n$ 行列で $B = (b_{jk})$ が $n\times p$ 行列の時， $A\cdot B$ または $AB$ の積を行列 $C = (c_{jk})$ と定義できます．ここで

$\displaystyle c_{jk} = \sum_{l = 1}^n a_{jl}b_{lk} \cdots (3)$

また $C$ は $m\times p$ 次です．

　一般に $AB \ne BA$ すなわち行列の積の交換法則は成り立たないことに注意してください．しかしながら結合法則と分配法則は成り立ちます，すなわち

$A(BC) = (AB)C,\ A(B + C) = AB + AC,\ (B + C)A = BA + CA \cdots (4)$

　ある行列 $A$ がそれ自身との積をつくれるのは正方行列の場合のみです．積 $A \cdot A$ は $A^2$ と記述します．同様に行列の累乗を定義できます．すなわち $A^3 = A\cdot A^2,\ A^4 = A\cdot A^3$ などです．

6. 行列の転置

　行列 $A$ の行と列を入れ替えることができるなら，その結果得られる行列は $A$ の転置と呼び， $A^T$ と記述します．記号では，仮に $A = (a_{jk})$ ならば $A^T = (a_{kj})$ と記述します．

以下を証明できます．

$(A + B)^T = A^T + B^T,\ (AB)^T = B^TA^T,\ (A^T)^T = A \cdots(5)$

7. 対称行列と歪対称行列

　ある正方行列 $A$ は $A^T = A$ の時対称と呼び， $A^T = - A$ の時 歪対称 と呼びます．

　任意の実正方行列（すなわち実数の要素のみからなる実正方行列）は常に実対称行列と実歪対称行列の和で表現できます．

8. 行列の複素共役

　仮に行列 $A$ のすべての要素 $a_{jk}$ が複素共役 $\bar{a}_{jk}$ で置き換えられたら，その結果得られた行列は $A$ の 複素共役 と呼び， $\bar{A}$ と記述します．

9. エルミート行列及び歪エルミート行列

　ある行列 $A$ がそれ自身の転置の複素共役に等しい時，すなわち $A = \bar{A}^T$ であるなら エルミート行列 と呼びます．仮に $A = -\bar{A}^T$ の場合は $A$ は 歪エルミート行列 と呼びます．仮に $A$ が実行列ならこれらは対称行列および歪対称行列にそれぞれ短縮されます．

10. 主対角線と行列のトレース

　仮に $A = (a_{jk})$ を正方行列とすると対角線上のすべての要素 $a_{jk}$ について $j = k$ であるものを principal あるいは 主対角線 と呼び，主対角線上の全要素の和を $A$ の トレース と呼びます．

　ある行列の $j \ne k$ なる要素について $a_{jk} = 0$ の時その行列を 対角行列 と呼びます．

11. 単位行列

　ある正方行列について主対角線上の全要素が 1 に等しく他の要素が全てゼロに等しい時 単位行列 と呼び， $I$ と記述します． $I$ の属性については非常に重要です．

$AI = IA = A,\ I^n = I,\ n = 1,2,3,\cdots(6)$

　単位ベクトルは行列代数において，普通の代数における数 1 と同じ役割を果たします．

12. 零行列またはヌル行列

　ある行列についてその要素が全てゼロに等しいなら ヌル行列 または 零行列 と呼び， $O$ または単に 0 と記述します．すべての行列 $A$ について同次数の 0 を考えると，

$A + 0 = 0 + A = A \cdots(7)$

　また仮に $A$ および 0 が正方行列なら

$A0 = 0A = 0 \cdots(8)$

　零行列は行列代数においては通常の対数における数 0 と同じ役割を果たします．

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21	22	23	24	25	26	27
28	29	30

日: 2014年4月18日

Some special definitions and operations involving matrices

1. Equality of Matrices

2. Addition of Matrices

3. Subtraction of Matrices

4. Multiplication of a Matrix by a Number

5. Multiplication of Matrices

6. Transpose of a Matrix

7. Symmetric and Skew-Symmetric matrices

8. Complex Conjugate of a Matrix

9. Hermitian and Skew-Hermitian Matrices

10. Principal Diagonal and Trace of a Matrix

11. Unit Matrix

12. Zero or Null matrix

いくつかの行列を含む特殊な定義と演算