A system of m×n numbers arranged in a rectangular formation along m rows and n columns and bounded by the brackets [ ] is called an m by n matrix; which is written as m × n matrix. A matrix is also denoted by a single capital letter A.
A = [aij], i to m, j = 1 to n.
1.2 Row Matrix:
A matrix with only one row is called a row matrix.
E.g. = [1 2 4]1x3
1.3 Column Matrix:
A matrix with only one column is called a column matrix.
1.4 Square Matrix:
A square matrix has an equal number of rows and columns. (i.e.) m = n.
1.5 Diagonal Matrix:
A square matrix is called a diagonal matrix if each of its non-diagonal elements are zero (i.e.) A square matrix A is diagonal if aij ≠ 0 for i = j and
aij = 0 for i ≠ j.
1.5.1. Properties of diagonal Matrix:
(a) diag [x, y, z] + diag [p, q, r] = diag [x + p, y + q, z + r]
(b) diag [x, y, z] × diag [p, q, r] = diag [xp, yq, zr]
(c) (diag [x, y, z])–1 = diag [1/x, 1/y/ 1/z]
(d) (diag[x, y, z])T = diag[x, y, z]
(e) diag [x, y, z]n = diag[xn, yn, zn]
(f) Eigen values of diag [x, y, z] = x, y and z.
(g) Determinant of diag [x, y, z] = | diag[x, y, z]| = xyz
1.6 Scalar Matrix:
A diagonal matrix whose diagonal elements are equal is called a scalar matrix.
1.7 Upper Triangular Matrix:
An upper triangular matrix is one whose entry values below the leading diagonal are zero.
1.8 Lower Triangular Matrix:
A lower triangular matrix is one whose entry values above the leading diagonal are zeros.
1.9 Equality of Matrices:
Two matrices are equal if they are of the same dimension and their corresponding entries are equal.
Eg: A = B
1.10 Identity Matrix:
An identity matrix is one whose leading diagonal entries are ones all the other entries are zero.
1.10.1. Properties of Identity Matrix:
(a) I is identity element for multiplication, so it is called multiplicative identity
(b) AI = IA = A
(c) In = I
(d) I–1 = I
(e) |I| = 1
1.11 Transpose of a Matrix:
A matrix obtained from any given matrix A, by interchanging its rows and columns is called the transpose of A an dis usually denoted by A' (or) AT.
Thus if A = [aij], then A' = [bij] where bij = aij.
1.11.1 Properties of Transpose of a Matrix:
If AT and BT be transpose of A and B respectively then,
- (AT)T = A
- (A + B)T = AT + BT
- (kA)T = kAT, k being any real number.
- (AB)T= BTAT
- (ABC)T = CTBTAT
1.12 Conjugate of a Matrix:
The matrix obtained from given matrix A, on replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A and denoted by A. Thus if A = [aij], then A = [bij] where bij = aij is called the conjugate of A.
1.13 Conjugate Transpose of A Matrix:
The conjugate of the transpose of a matrix A is called its conjugate transpose and denoted by AT. Thus AT.
1.14 Symmetric Matrices:
A square matrix A = [aij] is said to be symmetric if aij = aji ∀i, j.
(i.e.) the (i,j)th element is same as the (j, i)th element.
Therefore A = A'.
- If A and B and symmetric, then:
(a) A + B and A – B are also symmetric
(b) AB, BA may or may not be symmetric.
(c) Ak is symmetric when k is set of any natural number.
(d) AB + BA is symmetric.
(e) AB – BA is skew symmetric.
(f) A2, B2, A2 ± B2 are symmetric.
(g) KA is symmetric where k is any scalar quantity.
1.15 Skew Symmetric Matrices:
A square matrix A = [aij] is said to be skew symmetric of aij = -aji.
(i.e.) the (i, j)th element is the negative of its (j, i)th element.
Therefore A = -A'.
1.16 Hermitian Matrices:
A square matrix A = [aij] is said to be Hermitian if aij = aji.
(i.e.) the (i, j)th elements is the conjugate complex of the (j, i)th elements.
(i.e.) for a Hermitian matrix A, we have A = A'.
1.17 Skew-Hermitian Matrices:
A square matrix A = [aij] is said to be Skew-Hermitian if aij = -aji.
(i.e.) the (i, j)th elements is the negative conjugate complex of the (j, i)th elements.
1.18 Sub Matrix:
The matrix obtained on deleting any number of rows and columns of the given matrix A is called the sub-matrix of A.
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