Some special definitions and operations involving matrices

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1 1. Equality of Matrices
2 2. Addition of Matrices
3 3. Subtraction of Matrices
4 4. Multiplication of a Matrix by a Number
5 5. Multiplication of Matrices
6 6. Transpose of a Matrix
7 7. Symmetric and Skew-Symmetric matrices
8 8. Complex Conjugate of a Matrix
9 9. Hermitian and Skew-Hermitian Matrices
10 10. Principal Diagonal and Trace of a Matrix
11 11. Unit Matrix
12 12. Zero or Null matrix

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.