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Idempotent Matrix and Involutory Matrix

April 22, 2018

Idempotent Matrix

If a square matrix A is said to be an Idempotent matrix if $A^{2}=A$ .

Involutory Matrix
If a square matrix A is said to be an Involutory matrix if $A^{2}=I$ .

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Relation between Normal Matrix and Identity Matrix

Video Lecture We know that any number remains the same when multiplied by 1. e.g. : Similarly, any matrix remains the same when multiplied by the identity matrix. e.g. : ; where , A is a matrix I is a identity matrix Note : Identity matrix is a square matrix whose diagonal element is 1 and remaining element are zero's . e.g. :

Inverse Matrix and Invertible Matrix

Video Lecture If A is a square matrix of order m , and if there exists another square matrix B of the same order m , such that AB = BA = I , then B is called the inverse matrix of A and it is denoted by . In that case A is said to be invertible matrix. e.g. : Let Thus B is the inverse of A , in other words and A is inverse of B i.e.,

Matrix as a Sum of Symmetric Matrix and Skew-Symmetric Matrix

Video Lecture Every square matrix is uniquely expressible as the sum of a Symmetric matrix and a Skew- Symmetric matrix . Let A = P + Q , where P is Symmetric and Q is Skew-Symmetric. ∴ A ' = ( P + Q ) ' = P ' + Q ' = P + ( - Q ) = P - Q . solving , we get ∴ Representation of A as the sum

Idempotent Matrix and Involutory Matrix

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