Manuals/calci/SIGNATURE

MATRIX("SIGNATURE",order)

• $order$ 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 $\pm 1$
• So signature matrix is of the form:

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

• 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

See Also

• SHIFT
• CONFERENCE
• TRIANGULAR

References

• Signature Matrix