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.
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
|
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See Also
References