matrix analysis

A matrix is a rectangular array of elements arranged in horizontal rows and vertical columns, and usually enclosed in brackets. A matrix is real-valued (or, simply, real) if all its elements are real numbers or real-valued functions; it is complex-valued if at least one element is a complex number or a complex-valued function. If all its elements are numbers, then a matrix is called a constant matrix.

Matrices are designated by boldface uppercase letters. A general matrix A having r rows and ? columns may be written

where the elements of the matrix are double subscripted to denote location. By convention, the row index precedes the column index, thus, a₂₅ represents the element of Aappearing in the second row and fifth column, while a₃₁represents the element appearing in the third row and first column. A matrix Amay also be denoted as [a_ij], where a_ijdenotes the general element of A appearing in the ith row and jth column.

A matrix having r rows and ?columns has order (or size) "r by c," usually written r´c. Two matrices are equal if they have the same order and their corresponding elements are equal.

The transpose of a matrix A, denoted as A^T, is obtained by converting the rows of A into the columns of A one at a time in sequence. If A has order m´n, then A^T has order n´m.