Manuals/calci/SYMMETRIC

MATRIX("SYMMETRIC",order)

Description

This function returns the symmetric matrix of order 3.
A symmetric matrix is a square matrix that satisfies $A^{T}=A$ ,where $A^{T}$ denotes the transpose.
i.e., A square matrix which is equal to its transpose is called symmetric matrix.
So $a_{ij}=a_{ji}$ .
This also implies $A^{-1}A^{T}=I$ , 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

$A={\begin{pmatrix}43&-5&-93\\-5&-11&-75\\-93&-75&-7\\\end{pmatrix}}$

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.