 TOPICS # Eigen Decomposition Theorem

Let be a matrix of eigenvectors of a given square matrix and be a diagonal matrix with the corresponding eigenvalues on the diagonal. Then, as long as is a square matrix, can be written as an eigen decomposition where is a diagonal matrix. Furthermore, if is symmetric, then the columns of are orthogonal vectors.

If is not a square matrix (for example, the space of eigenvectors of is one-dimensional), then cannot have a matrix inverse and does not have an eigen decomposition. However, if is (with ), then can be written using a so-called singular value decomposition.

Eigen Decomposition, Eigenvalue, Eigenvector, Singular Value Decomposition

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## References

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. https://mathworld.wolfram.com/EigenDecompositionTheorem.html