Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew-symmetric matrices
A matrix is a set or group of numbers that are arranged in a rectangular array or a square enclosed in two brackets. A matrix is expressed by a bold capital letter and the elements are denoted by lower case letters e.g. matrix [A] with elements aij where ij define the order of the matrix.
Matrix algebra is used to reduce complicated systems of equations to simple expressions. It is adaptable to a systematic method of mathematical treatment and this type of algebra is well suited for computers. The term linear algebra is used to present the general theory of matrices and the algebraic operations associated with those matrices.The size of a matrix is defined by the number of rows and columns in the following way:
Types of matrices:
Column matrix or vector: The number of columns is always 1 and the number rows can be any integer.
Row matrix or vector: The number of rows is always 1 and the number columns can be any integer.
Rectangular matrix: In this type of matrix, the number of rows is not equal to the number of columns.
Square matrix: In this type of matrix, the number of rows is equal to the number of columns.The principal or main diagonal is composed of all elements aij for which i=j.
Diagonal matrix: In this matrix, all the elements except those on the main diagonal are zero.
Identity or unit matrix – I: It is a diagonal matrix with ‘1’ on the diagonal elements.
Zero (null) matrix – 0: All elements in this matrix are zero
Triangular matrix: This is a square matrix with element above or below the diagonal that is zero.
Upper triangular matrix: This is a triangular matrix whose elements below the main diagonal are all zero
Lower triangular matrix: This is a triangular matrix whose elements above the main diagonal are all zero
Scalar matrix: A scalar is a single constant number and a diagonal matrix with main diagonal elements equal to the same scalar makes up a scalar matrix.
Equality of matrices: Two matrices are equal to each other if all of the corresponding elements and dimensions are equal.
- IIfA = B, then B = A for all A and B
- IIfA = B, and B = C, then A = C for all A, B and C
The transpose of a matrix A is denoted as A’and is obtained by interchanging the rows and columns. Thus, if A has a size of m x n, A’ will have a size of n x m. When the transpose operation is applied two times, the original matrix is restored.
A matrix which is the same as its own transpose is called symmetric: AT = A
One which is the negative of its own transpose is called skew-symmetric: AT = -A
An orthogonal matrix has a transpose that is also its inverse: AT = A-1