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.

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Created: 2021-04-01 03:53:19, Last updated: 2021-04-01 03:53:19

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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

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