Matrix Eigenvalues and Eigenvectors

Matrix Eigenvalues and Eigenvectors - www.mathematics.me.uk...

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www.mathematics.me.uk Matrix Eigenvalues and Eigenvectors Eigenvalues and eigenvectors are the most fundamental characteristics of a square matrix 1 . For a square matrix A, the eigenvectors is the set of non-trivial (ie non-zero) vectors that are simply scaled when they are multiplied by A, with the scalings being equal to the eigenvalues . That is the eigenvalues and eigenvectors of a matrix A are the non-trivial vectors and scalars that satisfy the following equation: A x = x . Since x = I x , where I is the identity matrix then the equation can also be written in the alternative form A x - I x = 0 , or (A - I ) x = 0 . To find the eigenvalues, one method is to find the solutions to the equation =0. Where represents the determinant 2 . For a 2x2 matrix this usually involves the solution of a quadratic equation 3 . For each eigenvalue the corresponding eigenvector is found by finding a suitable x that satisfies the matrix-vector equation: (A - I ) x = 0 . Note that there is no unique solution to this equation; if x
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Matrix Eigenvalues and Eigenvectors - www.mathematics.me.uk...

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