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:
- A real square matrix is orthogonal if and only if its columns form an orthonormal basis on the Euclidean space ℝn, which is the case if and only if its rows form an orthonormal basis of ℝn.
- 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.)
- All orthogonal matrices are unitary (but the converse is not true.)
- Under the operation of multiplication, the n × n orthogonal matrices form the orthogonal group known as O(n).
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., .