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Math Help - eigenvalues and vectors related proof

  1. #1
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    eigenvalues and vectors related proof

    Hello!

    Could someone help me to proof one stuff?

    Say, I have A which is a matrix of a linear operator. λ in Spec(A) ( i.e. λ is eigenvalue of A ) x is an eigenvector for λ, and f(X) is a polynomial.

    How to proof that x is also an eigenvector for f(A)?

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by excellents View Post
    Hello!

    Could someone help me to proof one stuff?

    Say, I have A which is a matrix of a linear operator. λ in Spec(A) ( i.e. λ is eigenvalue of A ) x is an eigenvector for λ, and f(X) is a polynomial.

    How to proof that x is also an eigenvector for f(A)?

    Thanks in advance!
    Write the polynomial as [tex]f(A)]= a_nA^n+ a_{n-1}A^{n-1}+ \cdot\cdot\cdot+ a_1A+ a_nI[/itex]. Then f(A)x= aA^nx+ a_{n-1}A^{n-1}x+ \cdot\cdot\cdot+ a_1Ax+ a_nIx.

    Before you go further note that Ix= x, Ax= \lambda x, A^2x= A(\lambda x) = \lambda Ax= \lambda(\lambda x)= \lambda^2 x, etc.
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  3. #3
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    Where do A^n   x = \lambda^n    x come from? Is that Cayley–Hamilton theorem?
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  4. #4
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    Quote Originally Posted by excellents View Post
    Where do A^n x = \lambda^n x come from? Is that Cayley–Hamilton theorem?
    No, it is induction: it's easy to show that Ax=\lambda x \, \Longrightarrow A^2x=\lambda^2 x

    Now by induction show for any natural n.

    Tonio
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