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Symmetric Matrix

A square matrix in which aij = aji


Example :

A = Description: 498.png 
For symmetric matrix, AT = A

Skew Symmetric Matrix

A square matrix in which aij = – aji
All principal diagonal elements in skew symmetric matrix are zero.


Example :
A = Description: 503.png


In skew symmetric matrix, AT = – A

Hermitian Matrix


A square matrix in which aij is the conjugate complex of aji

i.e. Description: 508.png


Example :
Description: 513.png


a12 = 2 + 3ia21 conjugate complex of a12 = 2 – 3i

Skew Hermitian Matrix

A square matrix in which aij = Description: 518.png



Note : Principal diagonal elements of skew hermitian matrix should be zero or pure imaginary numbers.


Singular Matrix

A square matrix is a singular matrix if its determinant is zero. Otherwise, it is a non-singular matrix.

Orthogonal Matrix

A square matrix A is called an orthogonal matrix if AAT = ATA = I

Trace Of A Matrix

Let = [aij]n×n be a square matrix. Then the sum of all diagonal elements of A is called the trace of A and is denoted by tr (A). Thus,  tr(A) = Description: 523.png 

Idempotent Matrix

A square matrix A is called an idempotent matrix if A2 = A.

Properties of transpose of a matrix
Description: 528.png 
  • (A + B)T = AT + BT
  • (pA)T = pAT
  • (AB)T = BTAT (not ATBT)
  • AAT = ATA
Inverse of a Matrix
A–1 = Description: 553.png
Properties of Inverse of Matrix
  • (A–1)–1 = A
  • (AB)–1 = B–1A–1
  • (KA)–1 = K–1A–1 = Description: 573.png
  • | A–1 | = Description: 583.png
  • adj. (AB) = (adj. B) (adj. A)
  • | adj. A| = | A |n–1 where A is square matrix.

Echelon Matrix

A matrix is said to be echelon matrix if :
  1. Leading entry of each non zero row is 1
  2. The number of zeros proceeding this entry 1 is larger than number of proceeding rows.

Example :
A = Description: 599.png


Consistency of a Algebraic Equations

Note :
An equation is said to be homogenous if b = 0 i.e. of form AX = 0.
A system of equations is said to be consistent, if there are one or more solution for the equation.
A system of equations is said to be inconsistent, if there is no solution. When we reduce a matrix A to echelon form it is called argument matrix A*.
If p(A*) = p(A) the equations are consistent
If p(A*) ≠ p(A) the equations are inconsistent.
In a homogeneous system of equation (AX = 0),
If p(A) = no. of unknowns (n), it has no solution
If p(A) < no of unknown (n), it has infinite solutions.

Consider an augmented matrix AB which is reduced to echelon form.
If p(A) ≠ p(AB), system is inconsistent
If p(A) = p(AB) = no. of unknown (n), system is consistent with unique solution.
If p(A) = p(AB) < number of unknowns (n), system is consistent with infinite solutions.
Eigen Values and Eigen Vectors
For an n × n matrix A and a scalar λ, characteristic matrix of A is
A – λI. The equations | A – λI | = 0 is called characteristic equation and its roots is called Eigen values.

For Eigen values (λ) of matrix A, there is a non-zero vector X such that AX = λX is called characteristic vector (Eigen vector) of A.
Relations between Eigen values and Eigen vector
  • λ is a Eigen values of a matrix A if and only if there exists a non-zero vector X such that
  • AX = λ X
  • If X is a characteristic vector of a matrix A corresponding to the characteristic value λ, then KX is also a characteristic vector of A corresponding to the same characteristic value λ. Here K is any non-zero scalar.
  • If X is a Eigen vector of a matrix A, then X cannot correspond to more than one Eigen value of A.
  • Eigen vectors corresponding to distinct characteristic roots of a matrix are linearly independent.
Properties of Eigen Values
  • Sum of eigen values is equal to sum of element of principal diagonal of the matrix i.e.
  • Product of eigen values = determinant of the matrix.
  • If λ is eigen value of an orthogonal matrix, A then Description: 619.png is also its eigen value.
  • If λ is eigen value of matrix A, then Description: 629.png is eigen value of matrix A–1.
  • If λ1, λ2, λ3... are eigen values of matrix A, then λ1n, λ2n, λ3n ... are eigen values of matrix An.
  • Both A and AT will have same eigen values.
  • Eigen values of triangular matrix is equal to its diagonal elements.
  • Eigen values of real symmetric matrix are all real.
  • Eigen values of skew hermitian matrix are purely imaginary or zero.
  • It λ is eigen value of A, then Description: 665.png is eigen value of adj. (A).

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