Eigen Decomposition Theorem

Let P be a matrix of eigenvectors of a given square matrix A and D be a diagonal matrix with the corresponding eigenvalues on the diagonal. Then, as long as P is a square matrix, A can be written as an eigen decomposition


where D is a diagonal matrix. Furthermore, if A is symmetric, then the columns of P are orthogonal vectors.

If P is not a square matrix (for example, the space of eigenvectors of [1 1; 0 1] is one-dimensional), then P cannot have a matrix inverse and A does not have an eigen decomposition. However, if P is m×n (with m>n), then A can be written using a so-called singular value decomposition.

See also

Eigen Decomposition, Eigenvalue, Eigenvector, Singular Value Decomposition

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Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Singular Value Decomposition." §2.6 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 51-63, 1992.

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Eigen Decomposition Theorem

Cite this as:

Weisstein, Eric W. "Eigen Decomposition Theorem." From MathWorld--A Wolfram Web Resource.

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