# Linear algebra/Orthogonal matrix

A real-valued and square matrix is **orthogonal** (or **orthonormal**) if where is the transpose and is its inverse. Equivalently, is orthogonal when where is the identity matrix. It has a number of well-known properties:^{[1]}

- A real square matrix is orthogonal if and only if its columns form an orthonormal basis on the Euclidean space ℝ
^{n}, which is the case if and only if its rows form an orthonormal basis of ℝ^{n}.^{[1]} - The determinant of any orthogonal matrix is +1 or −1. But the converse is not true; having a determinant of ±1 is no guarantee of orthogonality. An orthogonal matrix with a determinant equal to +1 is called a special orthogonal matrix.
- An orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.
- All permutation matrices are orthogonal (but the converse is not true.)
^{[2]} - All orthogonal matrices are unitary (but the converse is not true.)
^{[3]} - Under the operation of multiplication, the
*n*×*n*orthogonal matrices form the orthogonal group known as O(*n*).

### ExampleEdit

Since orthogonal matrices form a group under multiplication, we can construct a non-trivial orthogonal matrix my multiplying two matrices that are easy to understand. Consider, for example,

Both and posses a symmetry that could lead one to postulate a non-existent symmetry among off-diagonal elements. But no such symmetry exists for . The upper-left 2x2 submatrix in represent a 45 degree rotation around the (or ) axis, plus and inversion through that same axis, i.e., .

## NotesEdit

- ↑
^{1.0}^{1.1}Most of this page is based on https://en.wikipedia.\org/w/index.php?title=Orthogonal_matrix&oldid=1028769520 - ↑ https://en.wikipedia.org/w/index.php?title=Permutation_matrix&oldid=1015641816#Properties
- ↑ https://en.wikipedia.org/w/index.php?title=Permutation_matrix&oldid=1015641816#Properties