Introduction to similar matrix and similar matrix transformation

How to perform similar matrices transformation?  Two square matrices A and B of the same order nxn are said to be similar if there exists an invertible matrix P such that B=P^-1AP. It can also be written as A=PBP^-1

The process of transforming a matrix A into another matrix B that is similar to it is called similar matrix transformation. 

The similar matrices have same characteristic equation that’s why their eigen values are always same. If x is an eigen vector of A and Y is the eigen vector of B, then the relation between their eigen vectors is:

Y=P^-1X

Prove that similar matrices have same eigen values

If B=P^-1AP

then

B-λI=P^-1AP-λI

As

P^-1P=I so put it in the above expression in place of I

B-λI=P^-1AP-λP^-1P

B-λI=P^-1(A-λI)P

Taking det on both sides, we get

det(B-λI)=det[P^-1(A-λI)P]

by the product rule of determinant, we get

det(B-λI)=det(P^-1)det(A-λI)det(P)

or

det(B-λI)=det(P^-1).det(P).det(A-λI)

or

det(B-λI)=det(P^-1P).det(A-λI)

since

P^-1P=I , so

det(B-λI)=det(I).det(A-λI)

and det(I)=1

so

det(B-λI)=det(A-λI)

Prove that similar matrices have same eigen vectors that are related as Y=P^-1X

Ax=λx

P^-1Ax=P^-1λx

or

P^-1Ax=λP^-1x

P^-1Ax=λP^-1Ix

P^-1AIx=λP^-1x

P^-1A(PP^-1)x=λP^-1x

or

(P^-1AP)P^-1x=λP^-1x

As

P^-1AP=B

so

BP^-1x=λP^-1x

call P^-1x=Y

so

BY=λY

where Y is the eigen vector of B corresponding to eigen value λ.

Application of similar matrix transformation

This transformation helps to find the higher powers of A. For example if A=A=PDP^-1

Then

A²=(PDP^-1)(PDP^-1)

or

A²=PD(P^-1P)DP^-1

A²=PD(I)DP^-1

A²=PD.DP^-1

A²=PD²P^-1

Similarly

A³=A².A

A³=(PD²P^-1).A

A³=(PD²P^-1).PDP^-1

A³=PD²⁽P^-1P)DP^-1

A³=PD²⁽I)DP^-1

A³=PD³P^-1

Similarly

A^K=PD^K P^-1

Example of similar matrices transformation

How to perform similar matrices transformation?

Eigen Values of similar Matrices

 

 

Example of similar matrices transformation

Also read here

https://eevibes.com/mathematics/linear-algebra/how-to-calculate-eigen-values-and-eigen-vectors/

what are the eigen values and eigen vectors? explain with examples