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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.
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).
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.
Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components.
The transpose of an upper triangular matrix is a lower triangular matrix and vice versa. A matrix which is both symmetric and triangular is diagonal. In a similar vein, a matrix which is both normal (meaning A * A = AA *, where A * is the conjugate transpose) and triangular is also diagonal.
One can always write = where V is a real orthogonal matrix, is the transpose of V, and S is a block upper triangular matrix called the real Schur form. The blocks on the diagonal of S are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived from complex conjugate eigenvalue pairs).
And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n / 2 of them.
Orthogonal matrix: A matrix whose inverse is equal to its transpose, A −1 = A T. They form the orthogonal group. Orthonormal matrix: A matrix whose columns are orthonormal vectors. Partially Isometric matrix: A matrix that is an isometry on the orthogonal complement of its kernel. Equivalently, a matrix that satisfies AA * A = A.