Study Notes on Basis for GATE 2018

Matrix and Dimensions

A matrix is a rectangular array of numbers arranged in m horizontal rows and n vertical columns. The numbers are called entries of the matrix and such a matrix is said to have dimension m x n. Matrices are usually denoted by capital letters. A, B …….

A = [aij], i to m, j = 1 to n.

or

Row Matrix:

A matrix with only one row is called a row matrix.

E.g. = [1 2 4]_1x3

Column Matrix:

A matrix with only one column is called a column matrix.

Square Matrix:

A square matrix has an equal number of rows and columns. (i.e.) m = n.

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.

Scalar Matrix:

A diagonal matrix whose diagonal elements are equal is called a scalar matrix.

Upper Triangular Matrix:

An upper triangular matrix is one whose entry values below the leading diagonal are zero.

Lower Triangular Matrix:

A lower triangular matrix is one whose entry values above the leading diagonal are zeros.

Equality of Matrices:

Two matrices are equal if they are of the same dimension and their corresponding entries are equal.

Eg: A = B

Where 

Identity Matrix:

An identity matrix is one whose leading diagonal entries are ones all the other entries are zero.

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) A^T.

Thus if A = [a_ij], then A' = [b_ij] where b_ij = a_ij.

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 = [a_ij], then A = [b_ij] where b_ij = a_ij is called the conjugate of A.

Eg: 

Conjugate Transpose of A Matrix:

The conjugate of the transpose of a matrix A is called its conjugate transpose and denoted by A^(H). Thus A^T.

Symmetric Matrices:

A square matrix A = [a_ij] is said to be symmetric if a_ij = a_ji ∀i, j.

(i.e.) the (i,j)^th element is same as the (j, i)^th element.

Therefore A = A'.

Skew Symmetric Matrices:

A square matrix A = [a_ij] is said to be skew symmetric of a_ij = -a_ji.

(i.e.) the (i, j)^th element is the negative of its (j, i)^th element.

Therefore A = -A'.

Hermitian Matrices:

A square matrix A = [a_ij] is said to be Hermitian if a_ij = a_ji.

(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'.

Skew-Hermitian Matrices:

A square matrix A = [a_ij] is said to be Skew-Hermitian if a_ij = -a_ji.

(i.e.) the (i, j)^th elements is the negative conjugate complex of the (j, i)^th elements.

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.

Determinant of a Matrix:

If A = [a_ij] be a square matrix of order n, then the determinant of [a_ij] of order n, is called the determinant of the square matrix A, denoted by |A|.

= 1(45-48) – 2(36-42) + 3(32-35)

= (-3) – 2(-6) + 3(-3)

= (-3) + 12 – 9

= -12 + 12

= 0.

|A| = 0.

Minors and Cofactors:

The determinant of a square submatrix of a matrix is called a minor of A.

Minors:

The minor of an element in a determinant is the determinant got by suppressing the row and column in which the element appears. The order of the minor of an element in a determinant A is one less than the order of |A|. The minor of the element in the i^th row. j^th column is denoted by m_ij [The determinant of a square sub matrix of a matrix A is called a minor of A].

Cofactor:

The signed minor is called cofactor. The cofactor of the element in the i^th row, j^th column is denoted by c_ij and c_ij = (-1)^i+j m_ij. The expansion of 3rd order determinant when expanded in terms of minors the sign of the cofactor of element will be as follows.

Adjoint or Adjugate of a Square Matrix:

The adjoint or adjugate of a square matrix A is a transpose of the matrix formed by cofactors of elements of |A| Adjoint of matrix. A is denoted as adj A

Note: If A is a non-singular square matrix then A(adj A) = (adj A) A = |A| I where I is unit matrix of same order as that of A.

Principle minor:

The principle minor of a matrix is the determinant of a square submatrix formed by deleting corresponding row and column vectors.

Eg: The principle minors of 3×3 matrix

order 1 and

of order 2.

Eigen Values and Eigen Vectors

A = [a_ij] be a square matrix of order n. if there exists a non-zero column vector X and a scalar λ such that AX = λX, then λ is called Eigen value of the matrix A and X is called an Eigen vector corresponding to the eigen value λ.

Characteristic Matrix:

The matrix A – λI is called characteristic matrix of given matrix A which is obtained by subtracting λ from diagonal elements of A.

Characteristic Polynomial:

The determinant |A – λI| when expanded will give a polynomial of degree n in λ which is called characteristic polynomial of matrix A.

Characteristic Equation:

The equation |A – λI| = 0 is called characteristic equation or secular equation of matrix A.

Characteristic Roots or Eigen values or Latent roots:

The roots of the characteristic eq. λ₁, λ₂ ……… λ_n are called characteristic roots or Eigen values or Latent roots.

Characteristic Vectors or Eigen Vectors:

Corresponding to each characteristic root λ there corresponds non-zero vector X satisfying the equation (A - λI) X = 0. The non-zero vectors X are called characteristic vectors or Eigen vectors.

Spectrum of a Matrix:

The set of all Eigen values of given matrix A is called the spectrum of A.