# Manuals/calci/SYMMETRIC

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**MATRIX("SYMMETRIC",order)**

- is the size of the Symmetric matrix.

## Description

- This function returns the symmetric matrix of order 3.
- A symmetric matrix is a square matrix that satisfies ,where denotes the transpose.
- i.e., A square matrix which is equal to its transpose is called symmetric matrix.
- So .
- This also implies , where I is the identity matrix.
- Because equal matrices have equal dimensions, only square matrices can be symmetric.
- An example for the symmetric matrix is

- The properties of symmetric matrices are:
- 1.Every square diagonal matrix is symmetric, since all off-diagonal entries are zero.
- 2.Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
- 3.Hermitian matrices are a useful generalization of symmetric matrices for complex matrices.
- In Calci, MATRIX("symmetric") gives the symmetric matrix with the integer numbers.
- The other way to give the syntax is MATRIX("symmetric:integer).
- The syntax is to get the positive numbers symmetric matrix is MATRIX("symmetric:positive integer").
- To get a negative numbers symmetric matrix is MATRIX("symmetric:negative integer").
- Also to get the symmetric matrix with the elements 0 and 1(boolean numbers) users give syntax as MATRIX("symmetric:boolean").
- So using Calci users can get a different types of symmetric matrices.

## Examples

- 1.MATRIX("symmetric") =84
- 2.MATRIX("symmetric",3)

-10 | 88 | 92 |

88 | 14 | -21 |

92 | -21 | -29 |

- 3.MATRIX("symmetric:boolean",4)

1 | 0 | 1 | 1 |

0 | 0 | 1 | 0 |

1 | 1 | 0 | 1 |

1 | 0 | 1 | 1 |

- 4.MATRIX("symmetric:integer",5)

-76 | -15 | 7 | -100 | -28 |

-15 | -32 | -98 | -100 | -87 |

7 | -98 | 47 | 52 | -72 |

-100 | -100 | 52 | -63 | 8 |

-28 | -87 | -72 | 8 | 76 |