In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express...

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...

In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the...

matrix U is special unitary if it is unitary and its matrix determinant equals 1. For real numbers, the analogue of a unitary matrix is an orthogonal...

B]\,.\end{aligned}}} The matrix exponential of a skew-symmetric matrix A {\displaystyle A} is then an orthogonal matrix R {\displaystyle R} : R = exp...

they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if RT = R−1 and det...

AA*). The determinant of any orthogonal matrix is either +1 or −1. A special orthogonal matrix is an orthogonal matrix with determinant +1. As a linear...

any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: For every real symmetric matrix A {\displaystyle...

orthogonal matrix is either +1 or −1. The special orthogonal group SO ⁡ ( n ) {\displaystyle \operatorname {SO} (n)} consists of the n × n orthogonal...

being normalized: a matrix that is slightly off may not be orthogonal any more and is harder to convert back to a proper orthogonal matrix. Quaternions also...

rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix R T...

−1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors...

case, "unitary" is the same as "orthogonal". Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as ⁠ A , {\displaystyle...

\phi \leq 2\pi .} Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue...

{\mathbf {A} }}.} A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if A T = A − 1 . {\displaystyle...

Symplectic vector space Symplectic group Symplectic representation Orthogonal matrix Unitary matrix Hamiltonian mechanics Linear complex structure Williamson theorem...

finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually...

Multilinear subspace learning Nonlinear dimensionality reduction Orthogonal matrix Signal separation Singular spectrum analysis Transform coding Varimax...

probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled...

projection matrix that is not an orthogonal projection matrix is called an oblique projection matrix. The eigenvalues of a projection matrix must be 0...

That is, for any k × k {\displaystyle k\times k} orthogonal matrix Q {\displaystyle Q} , the Gram matrix of Q v 1 , … , Q v n {\displaystyle Qv_{1},\dots...

\mathbb {R} ^{n}} and P {\displaystyle P} can be chosen to be an orthogonal matrix. For most practical work matrices are diagonalized numerically using...

Any real square matrix A may be decomposed as A = Q R , {\displaystyle A=QR,} where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning...

The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle...

that is it is an orthogonal matrix with determinant 1. That it is an orthogonal matrix means that its rows are a set of orthogonal unit vectors (so they...

post-multiplying an n-column matrix M, forming MP, permutes the columns of M. Every permutation matrix P is orthogonal, with its inverse equal to its...

the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = QTb = c, and...

A Householder matrix has eigenvalues ± 1 {\textstyle \pm 1} . To see this, notice that if x → {\textstyle {\vec {x}}} is orthogonal to the vector v...

covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the...

algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries...