Decomposition of a square matrix into symmetric and skew-symmetric matrices
This online calculator decomposes a square matrix into the sum of a symmetric and a skew-symmetric matrix.
This page exists due to the efforts of the following people:
 | Timur
|
This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/9233/. Also, please do not modify any references to the original work (if any) contained in this content.
The calculator below represents a given square matrix as the sum of a symmetric and a skew-symmetric matrix. You can find formulas and definitions below the calculator.
Decomposition of a square matrix into symmetric and skew-symmetric matrices
Symmetric matrix
A symmetric matrix is a square matrix those elements are symmetrical with respect to the main diagonal. That is, and
.
Skew-symmetric matrix
A skew-symmetric matrix is a square matrix, those elements are equal and negative with respect to the main diagonal. That is, and
.
Decomposition into symmetric and skew-symmetric
Every square matrix with entries from any field whose characteristic is different from 2 can uniquely be decomposed into the sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition.
Formula:
, where
- symmetric matrix
- skew-symmetric matrix
This formula is based on the fact that the sum A+AT is a symmetric matrix, the difference A-AT is a skew-symmetric matrix, and scalar multiplication retains these properties.
Comments