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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.

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.

PLANETCALC, Decomposition of a square matrix into symmetric and skew-symmetric matrices

Decomposition of a square matrix into symmetric and skew-symmetric matrices

Symmetric Matrix
 
Skew-symmetric matrix
 

Symmetric matrix

A symmetric matrix is a square matrix those elements are symmetrical with respect to the main diagonal. That is, \forall i,j:a_{{ij}}=a_{{ji}} and A=A^T.

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, \forall i,j:a_{{ij}}=-a_{{ji}} and A^T=-A.

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:
A = \frac {1}{2} (A+A^T) + \frac {1}{2} (A-A^T), where
\frac {1}{2} (A+A^T) - symmetric matrix
\frac {1}{2} (A-A^T) - 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.

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Creative Commons Attribution/Share-Alike License 3.0 (Unported) PLANETCALC, Decomposition of a square matrix into symmetric and skew-symmetric matrices

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