Matrix Definition. Main information
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Matrix definition
Definition.
Matrix with size n×m is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, which consisting of n rows and m columns.The number of rows and columns are defined the matrix size.
Matrix notation
Matrices are commonly written in box brackets or large parentheses:
| A = |  | 4 | 1 | -7 |  |
| -1 | 0 | 2 |
The matrix usually denoted by capital letters of the Latin alphabet. The matrix comprising n rows and m columns, called the matrix of size n×m. Also, the size of the matrix is written following way: An×m.
Matrix entries
Entries of the matrix A are denoted aij, where i is row number, in which is an element, j is the column number.
Example.
Entries of matrix A4×4:
| A = | | 4 | 1 | -7 | 2 | |
| -1 | 0 | 2 | 44 | |||
| 4 | 6 | 7 | 9 | |||
| 11 | 3 | 1 | 5 |
a11 = 4
Definition.
Matrix row is zero if all of its elements equal to zero.Definition.
If at least one of the elements of the matrix row is not zero, the row is called a non-zero.Example.
Zero and non-zero matrix row: | 4 | 1 | -7 | | < non-zero matrix row |
| 0 | 0 | 0 | |||
| 0 | 1 | 0 |
Definition.
Matrix column is zero if all of its elements equal to zero.Definition.
If at least one of the elements of the matrix column is not zero, the column is called a non-zero.Example.
Zero and non-zero matrix column: | 0 | 1 | -7 | |
| 0 | 0 | 2 | ||
^ | ^ | ^ |
non-non-zero matrix column
Matrix diagonal
Definition.
Main diagonal of matrix is the collection of entries aij where i = j.Definition.
Antidiagonal of matrix with size n×m is the collection of entries aij where i + j = n + 1.Example.
Main diagonal and antidiagonal of matrix: | 0 | 1 | -7 | | - main diagonalantidiagonal of matrix |
| 0 | 0 | 2 |
| 0 | 1 | -7 | | - main diagonalantidiagonal of matrix |
| 0 | 0 | 2 | |||
| 8 | 2 | 9 |
Definition.
The trace of square matrix A is defined to be the sum of the elements on the main diagonal.Definition.
The trace of matrix is denoted as tr(A) = a11 + a22 + ... + ann.
MatrixMatrix Definition. Main informationSystem of linear equations - matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix
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