# Symmetric matrix

This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.

View other properties of square matrices

## Contents

## Definition

### Verbal definition

A **symmetric matrix** is a square matrix that equals its own matrix transpose. Explicitly, a square matrix is termed a symmetric matrix if .

### Algebraic definition

Suppose is a positive integer and is a matrix. We say that is symmetric if the following holds:

Note that the condition is not required; the above definition is equivalent to:

Finally, because the equality condition is symmetric in , it suffices to check it for , so the above definition if equivalent to:

## Encoding

We can encode a symmetric matrix by simply storing the entries on or above the diagonal. This is a total of entries instead of the total of entries.

## Small cases

For the equal entries and , , we will use as the label. This is purely a matter of convention.

General description of symmetric matrix | Number of free parameters (equals ) | |
---|---|---|

1 | 1 | |

2 | 3 | |

3 | 6 |

## Matrix operations

### Invariant computation

For both the trace and the determinant, there does not seem to be a simpler way of computing them than the standard approach for square matrices.

### Operations

Matrix operation | Description in terms of encoding | Number of additions | Number of multiplications | Number of inverse computations | Set of symmetric matrices closed under operation? |
---|---|---|---|---|---|

Addition of two symmetric matrices | Add the corresponding entries of the two matrices | 0 | 0 | Yes | |

Multiplication of two symmetric matrices | Usual matrix multiplication algorithms; symmetry does not seem to offer speedup | for naive matrix multiplication | for naive matrix multiplication | 0 | No |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

scalar matrix | scalar multiple of identity matrix | Bisymmetric matrix, Diagonal matrix|FULL LIST, MORE INFO | ||

diagonal matrix | all off-diagonal entries are zero | | | ||

bisymmetric matrix | both symmetric and centrosymmetric | | | ||

symmetric Toeplitz matrix | Bisymmetric matrix|FULL LIST, MORE INFO |