# Orthogonal matrix

A real matrix is orthogonal (or, more precisely, orthonormal) when it has an inverse equal to its transpose[1][2]

PT=P-1

The term comes from the fact that the canonical orthonormal basis of the $\mathbb{R}^n\,$ is transformed by any orthonormal matrix (and only by orthonormal matrices) into another orthonormal basis.

Each orthonormal matrix represents one orthonormal basis, and reciprocally:

• If the columns of an orthonormal matrix are taken to be a set of vectors, then this set is an orthonormal basis
• If an orthonormal basis is written as the columns of a matrix, then this matrix is orthonormal [1]
• These properties are also valid for lines instead of columns

The concept generalizes to complex matrices as unitary matrices; however, in this case, instead of the the transpose it's necessary to use the conjugate-transpose.

## References

1. 1.0 1.1 Vecteurs et matrices, at http://benhur.teluq.uqam.ca, in French
2. Rowland, Todd. "Orthogonal Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.