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:

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 \begin{pmatrix} \pm & 0 & \cdots & 0 & 0 \\ 0 & \pm & \cdots & 0 & 0 \\ \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \pm & 0 \\ 0 & 0 & \cdots & 0 & \pm \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.