# Suppose that A is diagonalized by the matrix P and that the eigen-values of A are ......... Show that the eigenvalues of (A-`lambda1I```...

Suppose that * A *is diagonalized by the matrix

*and that the eigen-values of*

*P**are .........*

*A*Show that the eigenvalues of (A-`lambda1I```

`0,lambda2-lambda1,lambda3-lambda1,........................lambdan-lambda1`

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### 1 Answer

Matrix A is diagonalised by the matirx P ,then diagonal elements of matrix `P^(-1)AP` are eigen values of matrix A.Thus

`P^(-1)AP=[lambda_i]_(1<=i<=n)` ,is a diagonal matrix whose diagonal elements are `lambda_i ` ,

Let us consider matrix `(A-lambda_1I)` and diagonalise it,

`P^(-1)(A-lambda_1I)P=P^(-1)AP-P^(-1)(lambda_1I)P`

`=P^(-1)AP-lambda_1P^(-1)IP=[lambda_i]-lambda_1[1]`

`=[lambda_i]-[lambda_1]=[lambda_i-lambda_1]`

`` matrix multiplication is distributive ,also `lambda_i` 's are scalar quantities.

`P^(-1)(lambda_1I)P=lambda_1P^(-1)IP=lambda_1[1]=[lambda_1]`

Thus eigen values of `(A-lambda_1I)` are

`lambda_1-lambda_1,lambda_2-lambda_1,lambda_3-lambda_1,.......,lambda_n-lambda_1.`

`0,lambda_2-lambda_1,lambda_3-lambda_1,...............,lambda_n-lambda_1.` Hence proved