Manuals/calci/SYMMETRIC

MATRIX("SYMMETRIC",order)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^T=A} ,where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^T} denotes the transpose.
• i.e., A square matrix which is equal to its transpose is called symmetric matrix.
• So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}=a_{ji}} .
• This also implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.