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An orthogonal matrix Q is necessarily invertible (with inverse Q −1 = Q T), unitary (Q −1 = Q ∗), where Q ∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q ∗ Q = QQ ∗) over the real numbers. The determinant of any orthogonal matrix is either +1 or −1.
A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if =. A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, A is unitary if
In most cases the effect of the ambiguity is equivalent to the effect of a rotation matrix inversion (for these orthogonal matrices equivalently matrix transpose). Alias or alibi (passive or active) transformation The coordinates of a point P may change due to either a rotation of the coordinate system CS , or a rotation of the point P .
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.
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.
For a complex matrix, the transpose is replaced with the conjugate transpose. [12] ... If is an orthogonal projection matrix, that is, = ...
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: [ 3 ]
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).