Eigendecomposition, also called spectral decomposition, is one of the ways of decomposing a matrix. This decomposition is useful for analyzing properties of a matrix. This is similar to factoring a numbers into its prime factors. Some examples of the benefits of eigendecomposition are given below:

* If all eigenvalues are positive, then the matrix is positive definite. If all eigenvalues are positive or zero-valued, then the matrix is positive semi-definite. Similarly, for the definition of negative definite and negative semi-definite. There are benefits to knowing that a matrix is positive definite, positive semi-definite, negative definite and negative semi-definite.

* A matrix is singular if and only if any of the eigenvalues is zero

* To optimize quadratic expressions of the form $latex f = \mathbf{x}^T\mathbf{A}\mathbf{x}$, subject to $latex \|\mathbf{x}\| = 1. If $latex \mathbf{x}$ is an eigenvector of $latex \mathbf{A}$, then $latex f$ takes on the corresponding eigenvalue. Its maximum (or minimum) is the maximum (or minimum) eigevalue of $latex \mathbf{A}$.

* The determinant of a matrix is equal to the sum of all eigenvalues of a matrix.

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