I know that not all positive semi definite matrices have to be symmetric,and symmetric matrix have nice properties like positive eigen values and perpendicular eigen vectors.Eigendecomposition is applied on Symmetric matrix to decrease the computation time.Can we apply the same on positive semi definite?What is the exact use if a matrix is positive semidefinite? Is it only that it has a pseudo inverse and the function can be minimized easily?
2026-04-07 04:42:30.1775536950
Can we apply EigenDecomposition on Positive Semidefinite matrices?
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You can compute an Eigen-decomposition on all square matrices, not only symmetric matrices. But they are a set of matrices with special properties, that make EVs very applicable.
Symmetric matrices have the property of real eigenvalues (no imaginary part) but these do not need to be positive. A symmetric matrix with positive eigenvalues is called "positive definite" and if (at least) one eigenvalue is zero, it is called "positive semi-definite".
The use of an Eigenvalue decomposition depends on the area you are working in, as there are many different uses. A decomposition can be used to determine, if the matrix is in fact semi-definite. All negative (real parts of) Eigenvalues is needed for stable control systems, where matrices are not usually symmetric.