![]() For complex matrices, these results generalize in the obvious way: is the nearest Hermitian matrix to and is the nearest skew-Hermitian matrix to in any unitarily invariant norm. If we multiply a symmetric matrix by a scalar, the result. An example of non-uniqueness isĮntirely analogous to the above is the nearest skew-symmetric matrix problem, for which the solution is the skew-symmetric part for any unitarily invariant norm. Symmetric, diagonal, triangular matrices The sum of two symmetric matrices is a symmetric matrix. For the -norm, however, the nearest symmetric matrix is not unique in general. įor the Frobenius norm, is the unique nearest symmetric matrix, which follows from the fact that for symmetric and skew-symmetric. ![]() Where and are the symmetric part and the skew-symmetric part of, respectively, so the nearest symmetric matrix to is the symmetric part of. Simple examples of a matrix and a nearest symmetric matrix are A symmetric matrix can only be square and has the property aijaji. The most important examples of unitarily invariant norms are the -norm and the Frobenius norm. This symmetric matrix generator works entirely in your browser and is written in JavaScript. Such a norm depends only on the singular values of, and hence since and have the same singular values. A norm is unitarily invariant if for all unitary and. If the rows and columns of a matrix A are interchanged (so that. Fan and Hoffman (1955) showed that is a solution in any unitarily invariant norm. Note that the leading diagonal is a line of symmetry - a mirror line. Is this the best choice?Īs our criterion of optimality we take that is minimized over symmetric for some suitable norm. The natural way to symmetrize is to replace it by. In some cases, lack of symmetry is caused by rounding errors. For example, a finite difference approximation to a Hessian matrix can be nonsymmetric even though the Hessian is symmetric. ![]() What is the nearest symmetric matrix to a real, nonsymmetric square matrix ? This question arises whenever a symmetric matrix is approximated by formulas that do not respect the symmetry.
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