{"id":5156,"date":"2014-04-18T06:05:34","date_gmt":"2014-04-17T21:05:34","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5156"},"modified":"2014-08-01T18:35:51","modified_gmt":"2014-08-01T09:35:51","slug":"some-special-definitions-and-operations-involving-matrices","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5156","title":{"rendered":"Some special definitions and operations involving matrices"},"content":{"rendered":"
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1. Equality of Matrices<\/h4>\n

Two matrices A = (a_{jk}) and B = (b_{jk}) of the same order [i.e. equal numbers of rows and columns] are equal<\/em> if and only if a_{jk} = b_{jk}. <\/p>\n

2. Addition of Matrices<\/h4>\n

If A = (a_{jk}) and B = (b_{jk}) have the same order we define the sum<\/em> of A and B as A + B = (a_{jk} + b_{jk}) . <\/p>\n

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 <\/p>\n

A + B = B + A,\\ A + (B + C) = (A + B) + C \\cdots (2)<\/p>\n

3. Subtraction of Matrices<\/h4>\n

If A = (a_{jk}) , B = (b_{jk}) have the same order, we define the difference<\/em> of A and B as A - B = (a_{jk} - b_{jk}). <\/p>\n

4. Multiplication of a Matrix by a Number<\/h4>\n

If A = (a_{jk}) and \\lambda is any number or scalar, we define the product<\/em> of A by \\lambda as \\lambda A = A\\lambda = (\\lambda a_{jk}). <\/p>\n

5. Multiplication of Matrices<\/h4>\n

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<\/em> A\\cdot B or AB as the matrix C = (c_{jk}) where <\/p>\n

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

and where C is of order m\\times p. <\/p>\n

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. <\/p>\n

A(BC) = (AB)C,\\ A(B + C) = AB + AC,\\ (B + C)A = BA + CA \\cdots (4)<\/p>\n

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. <\/p>\n

6. Transpose of a Matrix<\/h4>\n

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

We can prove that <\/p>\n

(A + B)^T = A^T + B^T,\\ (AB)^T = B^TA^T,\\ (A^T)^T = A \\cdots(5) <\/p>\n

7. Symmetric and Skew-Symmetric matrices<\/h4>\n

A square matrix A is called symmetric<\/em> if A^T = A and skew-symmetric<\/em> if A^T = - A. <\/p>\n

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. <\/p>\n

8. Complex Conjugate of a Matrix<\/h4>\n

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<\/em> of A and is denoted by \\bar{A}. <\/p>\n

9. Hermitian and Skew-Hermitian Matrices<\/h4>\n

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<\/em>. If A = -\\bar{A}^T , then A is called skew-Hermitian<\/em>. If A is real these reduce to symmetric and skew-symmetric matrices respectively. <\/p>\n

10. Principal Diagonal and Trace of a Matrix<\/h4>\n

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<\/em> or main diagonal<\/em> and the sum of all elements is called trace<\/em> of A. <\/p>\n

A matrix for which a_{jk} = 0 when j \\ne k is called diagonal matrix<\/em>. <\/p>\n

11. Unit Matrix<\/h4>\n

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<\/em> and is denoted by I. An important property of I is that <\/p>\n

AI = IA = A,\\ I^n = I,\\ n = 1,2,3,\\cdots(6)<\/p>\n

The unit matrix plays a role in matrix algebra similar to that played by the number one in ordinary algebra. <\/p>\n

12. Zero or Null matrix<\/h4>\n

A matrix whose elements are all equal to zero is called the null<\/em> or zero matrix<\/em> and is often denoted by O or symply 0. For any matrix A having the same order as 0 we have <\/p>\n

A + 0 = 0 + A = A \\cdots(7)<\/p>\n

Also if A and 0 are square matrices, then<\/p>\n

A0 = 0A = 0 \\cdots(8)<\/p>\n

The zero matrix plays a role in matrix algebra similar to that played by the number zero of ordinary algebra. <\/p>\n<\/blockquote>\n