1. Eigenvalues Eigenvectors

# 1. Eigenvalues Eigenvectors - Eigenvalue problems Linear...

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Eigenvalue problems

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Linear transformation A x = λ x • A is a given square matrix • The scalar λ is unknown x is a vector to be determined Values of λ satisfying the equation A x = λ x are called eigenvalues and non trivial solutions x ( x # 0 ) are called eigenvectors .
Linear transformation A x = λ x We can rewrite the equation as (A- λ I) x = 0 The condition that there be a non trivial solution is det (A- λ I) = 0 For A(nxn), expansion of det gives a n th degree polynomial. The polynomial equation is called characteristic or secular equation. The n solutions of the secular equation are the eigenvalues of A.

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Eigenvalues and eigenvectors • Degenerate and non degenerate λ • A is singular if and only if one if its eigenvalues is zero • Sum of eigenvalues = Tr(A) • If ( x 1 , …x n ) are eigenvectors corresponding to distinct eigenvalues, then ( x 1 , …x n ) are linearly independent.
Eigenvalues and eigenvectors of Hermitian matrices A T = A in a real vector space, A is symmetric

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1. Eigenvalues Eigenvectors - Eigenvalue problems Linear...

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