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.
A symmetric matrix is a square matrix those elements are symmetrical with respect to the main diagonal. That is, and .
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.
- 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.