Matrix is an ordered rectangular array format in which numbers or functions takes place. array of n× m numbers existing in real or complex in the form of n horizontal lines called as rows and m vertical lines called as columns of matrix.
we represent matrices by the notation [ ] or ( ). numbers or functions which takes place called as the entries of Matix.
Representation of Matrices
An n × m matrix is generally represented as
In brief, the above matrix is represented by B= [bij] nxm. The number b11, b12, ….. b1m etc are known as the elements of the matrix B, where bij belongs to the ith row and jth column and is called the (i, j) th element of the given matrix B=[bij].
Types of Matrices
Column Matrix: A matrix is said to be a column matrix if and only if it contains only one column.
Row Matrix: A matrix is said to be a row matrix if and only if it contains only one row.
Symmetric Matrix: A square matrix A=[aij] is called a symmetric matrix if [aij] = [aji] for all i and j
that is transpose of A equal to itself.
what is the example of symmetric and skew-symmetric matrices?
what is the definition of symmetric and skew-symmetric matrices?
Skew-Symmetric Matrix: A square matrix A=[aij] is called a skew-symmetric matrix if [aij] = -[aji]
that is transpose of A is equal negative one time matrix A.
Square Matrix : A matrix in which contains equal number of rows and columns is said to be a square matrix. Thus an n× m matrix is said to be a square matrix if n = m and Matrix is known as a square matrix of order ‘m’.
Diagonal Matrix : A square matrix A = [aij]n×n is said to be a diagonal matrix if its all non diagonal elements are zero, that is a matrix A= [aij]n×n is called as diagonal matrix if
aij = 0, when i ≠ j.
i.e. diagonal matrix contains all nonzero elements its diagonally.
Identity Matrix : A square matrix in which matrix contains elements '1' at the diagonals and rest all elements are zeroes is called an identity matrix.
In other words, the square matrix B = [bij]n×n is an identity matrix, if
bij = 1, when i = j
bij = 0, when i ≠ j
Scalar Matrix : A diagonal matrix is called a scalar matrix if its diagonal elements are equal i.e. a square matrix A = [aij]n×n is said to be a scalar matrix if
aij = 0, when i ≠ j
aij = p, when i = j, for some constant p.
Hermitian Matrix :A square matrix is called as Hermitian matrix if A^θ=A, where A^θ represents transpose of conjugate matrix A.
Skew – Hermitian Matrix : A square matrix is called as skew-Hermitian matrix if A^θ = -A. where A^θ represents transpose of conjugate matrix A.
Orthogonal Matrix: A square matrix with Real entries whose rows and columns both are orthonormal vectors Or
(A)(At)=Identity matrix (I)=(At)(A)
where (At) represent transpose matrix of A
Null Matrix: A matrix is called as zero matrix or null matrix if all its containing elements are zero element. zero matrix is denoted by O.
Equal Matrix: Two matrices A = [aij] and B = [bij] are said to be equal if
1) each element of A is equal to the corresponding element of B i.e.
aij = bij for all i and j
2) Both are of the same order.
Nilpotent matrix: A matrix is called as involuntary matrix if its yields null matrix after multiplying with itself after finite time.
That is A^n=0 But A^(n-1)≠0 where n is an integer
Idempotent Matrix: A Matrix is called as Idempotent matrix if its yiels itself after multiplied by itself.
That is A^2=A
Involuntary matrix: A Matrix is called as Involuntary matrix if its yiels Identity matrix after multiplied by itself.
That is A^2=I implies that A= A^(-1) where A^(-1) denotes inverses of A
Upper triangular matrix: A square matrix which entries are zero below the principal (main) diagonal, called as Upper triangular matrix.
If aij be any arbitrary element of matrix then in upper triangular matrix aij =0 for all j < i
Lower triangular matrix : A square matrix which entries are zero above the principal (main) diagonal,called as lower triangular matrix.
If aij be any arbitrary element of matrix then in upper triangular matrix aij =0 for all i< j
Trace of Matrix
trace of any square matrix is the sum of all the entries on diagonal position in the matrix.
let A be any matrix then its trace is defined as
tr(A)=a11+a22+a33+a44+........+ann ,where aij are elements of the Matix A.
Properties of trace of matrix
suppose that A and B be two matrix having same order, then
1) tr(A+B) = tr(A) + tr(B)
2)tr(mA) = m tr(A)
3)tr(AB) = tr(BA) its true when both AB and BA are in existance