Difference between revisions of "Manuals/calci/SIGNATURE"

Revision as of 12:03, 4 May 2015

MATRIX("SIGNATURE",order)

This function returns the matrix of order 3 with the property of signature matrix.
A signature matrix is a diagonal elements are $\pm 1$
So signature matrix is of the form:

${\begin{pmatrix}\pm 1&0&\cdots &0&0\\0&\pm 1&\cdots &0&0\\\vdots &\ddots &\vdots \\0&0&\cdots &\pm 1&0\\0&0&\cdots &0&\pm 1\end{pmatrix}}$

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.

@@ Line 18: / Line 18: @@
 *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")
+{| class="wikitable"
+|-
+| 1 || 0 || 0
+|-
+| 0 || -1 || 0
+|-
+| 0 || 0 || 1
+|}
+*2.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==