Signature matrix

In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are either 1 or -1, that is, any matrix of the form

$${\displaystyle A={\begin{pmatrix}\pm 1&0&\cdots &0&0\\0&\pm 1&\cdots &0&0\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\cdots &\pm 1&0\\0&0&\cdots &0&\pm 1\end{pmatrix}}.}$$

Properties

Any signature matrix is its own inverse and hence is an involutory matrix. It is consequently a square root of the identity matrix.

Since signature matrices are both symmetric and involutory, they are orthogonal. Consequently, any linear transformation corresponding to a signature matrix constitutes an isometry. Geometrically, signature matrices represent a reflection in each of the axes corresponding to the negated rows or columns.

If ${\displaystyle A}$ is an ⁠${\displaystyle n\times n}$⁠ signature matrix, then:

• The determinant of ${\displaystyle A}$ is either 1 or -1 (since it is diagonal), and
• ${\displaystyle -n\leq \operatorname {tr} (A)\leq n}$ (since the diagonal values are either -1 or 1).

See also

• Metric signature
• Signature (matrix)

References

• Bapat, R. B. (2010). Graphs and Matrices. Universitext. London: Springer. ISBN 978-1-84882-980-0. MR 2797201.

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