Answer to Question #246368 in Linear Algebra for Trimmy

Question #246368

Show that if be the Eigenvalues of the matrix, then has the Eigenvalues

.

λ1, λ2, λ3, . . . λn A An

λn

1 , λ n

2 , λn

3 . . . λn

n


1
Expert's answer
2021-10-08T14:08:44-0400

Answer:-

We use the fact that characteristics polynomial of matrix A given by

P(X)=det(Aλ1)=(λ1X)(λ2X).....(λnX)P(X)=det(A-\lambda 1)=(\lambda 1-X)(\lambda 2-X).....(\lambda n-X)

We know that if A is n×n matrix and λi\lambda iiλiiλi is an Eigenvalue of A. It is a root of characteristics polynomial of A that is given by P(X)=det(AXl)=0P(X)=det(A-Xl)=0

In =n×n identity matrix

P(λ1)=0P(\lambda 1)=0

If λ1,λ2.......λn\lambda 1, \lambda 2.......\lambda n are Eigenvalue of matrix A then

P(λ1)=0P(\lambda 1)=0 which is a monic polynomial

P(X)=(λ1X)(λ2X)........(λnX)det(AXIn)=(λ1X)(λ2X)........(λnX)P(X)=(\lambda 1 -X)(\lambda 2-X)........(\lambda n-X)\\ det(A-XIn)=(\lambda 1 -X)(\lambda 2-X)........(\lambda n-X)

Put X=0

We get

det(A)=(λ10)(λ20).......(λn0)det(A)=(\lambda 1 -0)(\lambda 2-0).......(\lambda n-0)

=λ1,λ2......λn=\lambda 1,\lambda 2......\lambda n

det(A)=(λ1,λ2)......(λn)det(A)=(\lambda 1,\lambda 2)......(\lambda n)

Hence proved.


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