Manuals/calci/SIGNATURE
MATRIX("SIGNATURE",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 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 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 \pm 1}
- So signature matrix is of the form:
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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.
Examples
- 1. MATRIX("signature")
| 1 | 0 | 0 |
| 0 | -1 | 0 |
| 0 | 0 | 1 |
- 2.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 |