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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.
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.
Thus a real symmetric matrix A can be decomposed as =, where Q is an orthogonal matrix whose columns are the real, orthonormal eigenvectors of A, and Λ is a diagonal matrix whose entries are the eigenvalues of A.
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).
For a symmetric matrix A, upon convergence, AQ = QΛ, where Λ is the diagonal matrix of eigenvalues to which A converged, and where Q is a composite of all the orthogonal similarity transforms required to get there. Thus the columns of Q are the eigenvectors.
A matrix = [] belongs to the orthogonal group if AQA T = Q, that is, a 2 – ωb 2 = 1, ac – ωbd = 0, and c 2 – ωd 2 = –ω. As a and b cannot be both zero (because of the first equation), the second equation implies the existence of ε in F q , such that c = εωb and d = εa .
Nilpotent matrix: A square matrix satisfying A q = 0 for some positive integer q. Equivalently, the only eigenvalue of A is 0. Normal matrix: A square matrix that commutes with its conjugate transpose: AA ∗ = A ∗ A: They are the matrices to which the spectral theorem applies. Orthogonal matrix: A matrix whose inverse is equal to its ...
The indefinite special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1. Unlike in the definite case, SO(p, q) is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO + (p, q) and O + (p, q), which has 2 components – see ...