linear algebra, an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates...

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...

normal form Weight module – associative algebra generalization Orthogonal diagonalization Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second...

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension...

In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission used in digital modulation for encoding digital...

a right angle, whereas orthogonal is used in generalizations, such as orthogonal vectors or orthogonal curves. Orthogonality is also used with various...

what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation...

form, there is an orthogonal diagonalization; that is, an orthogonal change of variables that puts the quadratic form in a "diagonal form" λ 1 x ~ 1 2...

elements of the algebra are real numbers. This basis may be found by orthogonal diagonalization. The free algebra generated by V may be written as the tensor...

combinatorics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all...

always has an orthogonal basis. This can be proven by induction. A basis C is orthogonal if and only if the matrix representation A is a diagonal matrix. In...

of matrix diagonalization, where one tries to find a suitable coordinate system in which the matrix of a linear transformation is diagonal. The first...

of a tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form...

In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave...

definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean...

In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots...

can be diagonalized by an orthogonal matrix. More explicitly: For every real symmetric matrix A {\displaystyle A} there exists a real orthogonal matrix...

with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices. Generators of rotations...

with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices. More intrinsically...

right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these "cuboids", without qualifying...

vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line...

matrices and D is a diagonal matrix. The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable...

(The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus A = U D U ∗ {\displaystyle A=UDU^{*}} and A ∗ = U D...

^{\mathsf {T}}A\mathbf {y} .} An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal...

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations...

system involving orthogonals (diagonal lines that meet at a central vanishing point) and transversals (the lines crossing these orthogonals) which work together...

sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of orthogonal projection operators. The...

K such that k(X) is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies...

acts as a traditional inverse of ⁠ A {\displaystyle A} ⁠ on the subspace orthogonal to the kernel. In the following discussion, the following conventions...

three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components {vk} (the three mutually perpendicular sides):...