# 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.

#### 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.

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