Difference between revisions of "Manuals/calci/SIGNATURE"
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| − | <div style="font-size:30px">'''SIGNATURE'''</div><br/> | + | <div style="font-size:30px">'''MATRIX("SIGNATURE",order)'''</div><br/> |
| + | *<math>order</math> is the size of the Signature matrix. | ||
| + | |||
| + | ==Description== | ||
| + | *This function returns the matrix of order 3 with the property of signature matrix. | ||
| + | *A signature matrix is a diagonal elements are <math>\pm 1</math> | ||
| + | *So signature matrix is of the form: | ||
| + | <math>\begin{pmatrix} | ||
| + | \pm 1 & 0 & \cdots & 0 & 0 \\ | ||
| + | 0 & \pm 1 & \cdots & 0 & 0 \\ | ||
| + | \vdots & \vdots &\ddots & \vdots & \vdots \\ | ||
| + | 0 & 0 & \cdots & \pm 1 & 0 \\ | ||
| + | 0 & 0 & \cdots & 0 & \pm 1 | ||
| + | \end{pmatrix}</math> | ||
| + | *Any such matrix is its own inverse, hence is an involutory matrix. | ||
| + | *It is consequently a square root of the identity matrix. | ||
| + | *Also that not all square roots of the identity are signature matrices. | ||
| + | *The signature matrices are both symmetric and involutory,i.e.,they are orthogonal. | ||
| + | *Consequently, any linear transformation corresponding to a signature matrix constitutes an isometry. | ||
| + | |||
| + | ==Examples== | ||
| + | *1. MATRIX("signature")= 1 | ||
| + | *2.MATRIX("signature",3) | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | | 1 || 0 || 0 | ||
| + | |- | ||
| + | | 0 || -1 || 0 | ||
| + | |- | ||
| + | | 0 || 0 || 1 | ||
| + | |} | ||
| + | *3.MATRIX("signature",6) | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | | 1 || 0 || 0 || 0 || 0 || 0 | ||
| + | |- | ||
| + | | 0 || -1 || 0 || 0 || 0 || 0 | ||
| + | |- | ||
| + | | 0 || 0 || 1 || 0 || 0 || 0 | ||
| + | |- | ||
| + | | 0 || 0 || 0 || 1 || 0 || 0 | ||
| + | |- | ||
| + | | 0 || 0 || 0 || 0 || -1 || 0 | ||
| + | |- | ||
| + | | 0 || 0 || 0 || 0 || 0 || 1 | ||
| + | |} | ||
| + | |||
| + | ==See Also== | ||
| + | *[[Manuals/calci/SHIFT| SHIFT]] | ||
| + | *[[Manuals/calci/CONFERENCE| CONFERENCE]] | ||
| + | *[[Manuals/calci/TRIANGULAR| TRIANGULAR]] | ||
| + | |||
| + | ==References== | ||
| + | *[http://en.wikipedia.org/wiki/Signature_matrix Signature Matrix] | ||
Latest revision as of 02:40, 26 October 2015
MATRIX("SIGNATURE",order)
- is the size of the Signature matrix.
is the size of the Signature matrix.Description
- This function returns the matrix of order 3 with the property of signature matrix.
- A signature matrix is a diagonal elements are
- So signature matrix is of the form:
- Any such matrix is its own inverse, hence is an involutory matrix.
- It is consequently a square root of the identity matrix.
- Also that not all square roots of the identity are signature matrices.
- The signature matrices are both symmetric and involutory,i.e.,they are orthogonal.
- Consequently, any linear transformation corresponding to a signature matrix constitutes an isometry.
Examples
- 1. MATRIX("signature")= 1
- 2.MATRIX("signature",3)
| 1 | 0 | 0 |
| 0 | -1 | 0 |
| 0 | 0 | 1 |
- 3.MATRIX("signature",6)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |