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# Eigen value - Definition Let Ann be a matrix If there...

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Definition: Let n n A × be a matrix. If there exists a nonzero vector n R x such that x x λ = A , then real no. λ is called an eigenvalue of A and every x is called an eigenvector of A associated with λ .

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Eigenvalues are also called proper values, characteristic values and latent values ; and eigenvectors are also called proper vectors, characteristic vectors and latent vectors .
Remarks : 1. 0 cannot be an eigenvector but 0 can be an eigenvalue. 2. A may have many different eigenvectors associated with an eigenvalue λ , since 0 , r r x is also an eigenvector if x is an eigenvector of A .

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Ex.1 : Let n I A = . Find it’s eigenvalues and corresponding eigenvectors. Sol. : Eigenvalue : 1 = λ Eigenvector : every nonzero vector n R x
Ex.2 : Let = 2 6 1 3 A Find the eigenvalues of A and associated eigenvectors.

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Sol. : Eigenvalue : 5 , 0 = λ Eigenvectors associated with 0 = λ : 0 , 3 - r r r , In particular, - 3 1 Eigenvectors associated with 5 = λ : 0 , 2 r r r , In particular, 2 1
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Eigen value - Definition Let Ann be a matrix If there...

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