Algebra of Matrices
In this chapter, we shall discuss matrix algebra and its use in solving linear system of algebraic equations AX = B and in solving the Eigen value problem AX = λX.
Matrix Algebra
Definition of Matrix
A system of m × n numbers arranged in the form of a rectangular array having m rows and n columns is called an matrix of order m × n.
If A = [aij ]m x n be any matrix of order m × n then it is written in the form:
Horizontal lines are called rows and vertical lines are called columns.
Special Types of Matrices
1. Square Matrix: An m × n matrix for which m = n (The number of rows is equal to number of columns) is called square matrix. It is also called an n-rowed square matrix. i.e. A = [aij ]n × n. The elements aij ⏐ i = j, i.e. a11, a22…. are called DIAGONAL ELEMENTS and the line along which they lie is called PRINCIPLE DIAGONAL of matrix. Elements other than a11, a22, etc are called off-diagonal elements i.e. aij ⏐ i ≠ j
2. Scalar Matrix: A scalar matrix is a diagonal matrix with all diagonal elements being equal.
3. Unit Matrix or Identity Matrix: A square matrix each of whose diagonal elements is 1 and each of whose non-diagonal elements are zero is called unit matrix or an identity matrix which is denoted by I. Identity matrix is always square.
4. Null Matrix: The m × n matrix whose elements are all zero is called null matrix. Null matrix is denoted by O. Null matrix need not be square. aij = 0 ∀ i, j
5. Upper Triangular Matrix: An upper triangular matrix is a square matrix whose lower off-diagonal elements are zero, i.e. aij = 0 whenever i > j
It is denoted by U.
6. Lower Triangular Matrix: A lower triangular matrix is a square matrix whose upper off-diagonal triangular elements are zero, i.e. aij = 0 whenever i < j. The diagonal and lower off-diagonal elements may or may not be zero. . {aij = 0 if i< j
It is denoted by L,
7. Idempotent Matrix: A matrix A is called Idempotent if A2 = A.
8. Involuntary Matrix: A matrix A is called Involutory if A2 = I.
9. Nilpotent Matrix: A matrix A is said to be nilpotent of class x or index x if Ax = O and Ax – 1 ≠ O i.e. x is the smallest index which makes Ax = O.
10. Singular matrix: If the determinant of a matrix is zero, then matrix is called as singular matrix.
11. Row Matrix: A matrix with only one row is called as Row Matrix. A row matrix can have any number of columns i.e., it has an order 1 × n where n ∈ natural number.
12. Column Matrix: A matrix with only one column is called as column matrix. A column matrix can have any number of rows i.e., it has an order m × 1 where m ∈ natural number.
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