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# 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^{-1}AP. 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^{-1}X

## Prove that similar matrices have same eigen values

If B=P^{-1}AP

then

B-λI=P^{-1}AP-λI

As

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

B-λI=P^{-1}AP-λP^{-1}P

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^{-1}P).det(A-λI)

since

P^{-1}P=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^{-1}X

Ax=λx

P^{-1}Ax=P^{-1}λx

or

P^{-1}Ax=λP^{-1}x

P^{-1}Ax=λP^{-1}Ix

P^{-1}AIx=λP^{-1}x

P^{-1}A(PP^{-1})x=λP^{-1}x

or

(P^{-1}AP)P^{-1}x=λP^{-1}x

As

P^{-1}AP=B

so

BP^{-1}x=λP^{-1}x

call P^{-1}x=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

A^{2}=(PDP^{-1})(PDP^{-1})

or

A^{2}=PD(P^{-1}P)DP^{-1}

A^{2}=PD(I)DP^{-1}

A^{2}=PD.DP^{-1}

A^{2}=PD^{2}P^{-1}

Similarly

A^{3}=A^{2}.A

A^{3}=(PD^{2}P^{-1}).A

A^{3}=(PD^{2}P^{-1}).PDP^{-1}

A^{3}=PD^{2(}P^{-1}P)DP^{-1}

A^{3}=PD^{2(}I)DP^{-1}

A^{3}=PD^{3}P^{-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