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Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA T will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.
More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if R T = R −1 and det R = 1. The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO( n ) , one example of which is ...
A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix R, note that det R T = det R implies (det R) 2 = 1, so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3).
A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set (or orthogonal system). If the vectors are normalized, they form an orthonormal system. An orthogonal matrix is a matrix whose column vectors are orthonormal to each other.
A set of three 2 × 2 complex Hermitian and unitary matrices. When combined with the I 2 identity matrix, they form an orthogonal basis for the 2 × 2 complex Hermitian matrices. Redheffer matrix: Encodes a Dirichlet convolution. Matrix entries are given by the divisor function; entires of the inverse are given by the Möbius function.
The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices. QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column.
Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation. If R has more than one invariant vector then φ = 0 and R = I. Any vector is an invariant vector of I.
In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V.