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

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

1 3 8 5 4 1 2 3 0

Square matrix

Calculation precision

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Rounded

Digits after the decimal point:

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Symmetric Matrix

 

Skew-symmetric matrix

 

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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+A^T is a symmetric matrix, the difference A-A^T is a skew-symmetric matrix, and scalar multiplication retains these properties.