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Method in linear algebra
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
Orthogonal_diagonalization
Matrices similar to diagonal matrices
spaces. Additional geometric visualizations of orthogonal diagonalization, including reflection and orthogonal projection matrices, are available at Wikimedia
Diagonalizable_matrix
Various meanings of the terms
Orthogonality is a term with various meanings depending on the context. In mathematics, orthogonality is the generalization of the geometric notion of
Orthogonality
Type of group in mathematics
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension
Orthogonal_group
Real square matrix whose columns and rows are orthogonal unit vectors
In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way
Orthogonal_matrix
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
symmetric three-by-three matrix. This symmetric matrix can be orthogonally diagonalized by a set of mutually perpendicular principal axes. Torques around
Moment_of_inertia
Polynomial with all terms of degree two
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
Quadratic_form
Property of a mathematical matrix
what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation
Definite_matrix
Mathematical problem
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
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Vector space with generalized dot product
definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean
Inner_product_space
Algebra based on a vector space with a quadratic form
of the algebra are real numbers. Such a basis may be found by orthogonal diagonalization. The free algebra generated by V may be written as the tensor
Clifford_algebra
Concept in mathematics
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
Symmetric_bilinear_form
Principle in geometry and linear algebra
of matrix diagonalization, where one tries to find a suitable coordinate system in which the matrix of a linear transformation is diagonal. The first
Principal_axis_theorem
Method of encoding digital data on multiple carrier frequencies
In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission used in digital modulation for encoding digital
Orthogonal frequency-division multiplexing
Orthogonal_frequency-division_multiplexing
Set of polynomials where any two are orthogonal to each other
mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other
Orthogonal_polynomials
Most widely known generalized inverse of a matrix
acts as a traditional inverse of A {\displaystyle A} on the subspace orthogonal to the kernel. In the following discussion, the following conventions
Moore–Penrose_inverse
Matrix with nonzero elements on the main diagonal and the diagonals above and below it
of a tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form
Tridiagonal_matrix
Form of a matrix
with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices. More intrinsically
Skew-symmetric_matrix
Special quadrilateral whose diagonals intersect at right angles
the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular)
Orthodiagonal_quadrilateral
Cuboid with all right angles and equal opposite faces
right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these "cuboids", without qualifying
Rectangular_cuboid
Variants of chess with multiple boards at different levels
unicorns; pawns can move and capture one cell forward, either orthogonally, diagonally, or triagonally Tamerlane Cubic Chess – adapting Tamerlane chess
Three-dimensional_chess
Matrix with the same number of rows and columns
^{\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
Square_matrix
Set of coordinates where the coordinate hypersurfaces all meet at right angles
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
Orthogonal_coordinates
Type of algorithm
V^{\mathsf {T}}} where U and V are orthogonal and Σ {\displaystyle \Sigma } is diagonal. Next, record if the orthogonal matrices contain a reflection, d
Kabsch_algorithm
Sculpture by Donatello
system involving orthogonals (diagonal lines that meet at a central vanishing point) and transversals (the lines crossing these orthogonals) which work together
The Feast of Herod (Donatello)
The_Feast_of_Herod_(Donatello)
Non-associative algebras with positive-definite quadratic form
K such that k(X) is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Polynomial sequence
mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α) n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function
Gegenbauer_polynomials
Five-pointed star polygon
stellated dodecahedron Small ditrigonal icosidodecahedron Dodecadodecahedron Orthogonal projections of higher dimensional polytopes can also create pentagrammic
Pentagram
Matrix equal to its transpose
can be diagonalized by an orthogonal matrix. More explicitly: For every real symmetric matrix A {\displaystyle A} , there exists a real orthogonal matrix
Symmetric_matrix
Solid with six equal square faces
vertices, edges, and faces. All three square faces surrounding a vertex are orthogonal to each other, meaning the planes are perpendicular, forming a right angle
Cube
Orthogonal group of an indefinite quadratic form
In mathematics, the indefinite orthogonal group, O ( p , q ) {\displaystyle \operatorname {O} (p,q)} is the Lie group of all linear transformations of
Indefinite_orthogonal_group
Matrix decomposition
M ∗ M {\displaystyle \mathbf {M} ^{*}\mathbf {M} } . Applying the diagonalization result, the unitary image of its positive square root T f {\displaystyle
Singular_value_decomposition
36×36 grid variant of Japanese chess
direction, orthogonal or diagonal, except orthogonally forward. Notation: FbWsW Left general (左将) Step: The left general can move one square orthogonally vertically
Taikyoku_shogi
difference operators diagonalized by them. Furthermore, there is a large class of interesting families of multivariable orthogonal polynomials associated
Koornwinder_polynomials
Relationship between two lines that meet at a right angle
instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced
Perpendicular
Matrix equal to its conjugate-transpose
{\displaystyle n} linearly independent eigenvectors; A {\displaystyle A} has orthogonal eigenvectors for distinct eigenvalues; even if A {\displaystyle A}
Hermitian_matrix
Catalan solid with 30 faces
Their boundless faces and edges as elongated prisms or pyramids are orthogonal to the central planes and faces of their dual hemipolyhedra; the coincidental
Rhombic_triacontahedron
Flat-sided three-dimensional shape
be constructed with integers coordinates. Orthogonal polyhedra are polyhedra where all edges are orthogonal, parallel to all three axes of Cartesian coordinate
Polyhedron
namely, a non-square matrix with orthogonal columns/rows. All real orthogonal matrices and real invertible diagonal matrices, for instances, are G-matrices
G-matrix
Array of numbers
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
Matrix_(mathematics)
Generalization of Lie groups
In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations
Schur_orthogonality_relations
Large board variant of shogi
move two squares orthogonally or range forward diagonally (fBR2). In the SRZ, it moves one square orthogonally or two squares diagonally (WB2). Japanese
Dai_dai_shogi
Random matrix with gaussian entries
both mathematics and physics. The three main examples are the Gaussian orthogonal (GOE), unitary (GUE), and symplectic (GSE) ensembles. These are classified
Gaussian_ensemble
Concepts from linear algebra
vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization. In essence, an eigenvector v of a linear transformation T is a nonzero
Eigenvalues_and_eigenvectors
Relation between sides of a right triangle
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):
Pythagorean_theorem
1966 mathematics textbook by Serge Lang
introduces polynomials briefly, and chapter ten covers triangulation, diagonalization and the Hamilton-Cayley theorem. Chapter eleven introduces the polynomial
Linear_Algebra_(book)
Result about when a matrix can be diagonalized
simpler computations involving the corresponding diagonal matrix of eigenvalues. The concept of diagonalization is relatively straightforward for operators
Spectral_theorem
Method of data analysis
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
Principal_component_analysis
Square tiles used in graphic design
Stanley Smith. The tiles originally studied by Truchet are split along the diagonal into two triangles of contrasting colors. The tiles have four possible
Truchet_tiling
Matrix decomposition
\left[{\begin{smallmatrix}1&0\\1&3\end{smallmatrix}}\right]} cannot be diagonalized in an orthogonal basis (see Example below). One special case is when A is a normal
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Independent component analysis algorithm
Joint Approximation Diagonalization of Eigen-matrices (JADE) is an algorithm for independent component analysis that separates observed mixed signals
Joint Approximation Diagonalization of Eigen-matrices
Joint_Approximation_Diagonalization_of_Eigen-matrices
Mathematical operation in quantum optics, general relativity and other areas of physics
correspond to the orthogonal symplectic transformations (i.e., rotations) and the squeezing factor r {\displaystyle r} corresponds to the diagonal transformation
Bogoliubov_transformation
Matrix that commutes with its conjugate transpose
(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
Normal_matrix
Orthonormalization of a set of vectors
S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}} for k ≤ n and generates an orthogonal set S ′ = { u 1 , … , u k } {\displaystyle S'=\{\mathbf {u} _{1},\ldots
Gram–Schmidt_process
Matrix factorisation in mathematics
simultaneously brought to quasi-triangular form by an orthogonal matrix. There exists an orthogonal matrix Q such that, for every Ai in the given family
Schur_decomposition
Ancient Indian strategy board game
initial orthogonal step, also without the ability to jump over an intervening piece or pawn. One step straight forward or one step in any diagonal direction
Chaturanga
Mathematical operation on vectors in 3D space
The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a
Cross_product
Bidiagonalization is one of unitary (orthogonal) matrix decompositions such that U* A V = B, where U and V are unitary (orthogonal) matrices; * denotes Hermitian
Bidiagonalization
English chess variant inventor (1897–1974)
Raumschach, except that pawns move and capture one step forward (either orthogonally, diagonally, or vertexally), but not directly upward or downward. As in chess
V._R._Parton
American activist
combinatorialists as E.T. Parker and Walter Wallis, a construction of orthogonal pairs of doubly diagonal Latin squares of order 10, thus completing the proof that
Fred_Cherry
Coordinate system using perpendicular axes
origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called
Cartesian_coordinate_system
Type of matrix
with the above-mentioned block-diagonalization for skew-symmetric matrices, implies the block-diagonalization for orthogonal matrices. Generators of rotations
Infinitesimal_rotation_matrix
Visual illusion
the perceived direction of movement is usually orthogonal to the orientation of the stripes (diagonal, in this case). The perceived direction of movement
Barberpole_illusion
Playing piece with non-standard chess rules
different pieces. Directions are defined as orthogonal (i.e. horizontal or vertical), like a rook; diagonal, like a bishop; and hippogonal, like a knight
Fairy_chess_piece
Central object in linear algebra; mapping vectors to vectors
eigenvectors are derived from it via the characteristic polynomial. With diagonalization, it is often possible to translate to and from eigenbases. Most common
Transformation_matrix
Representation of a matrix as a product
floating point. Similarly, 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
Matrix_decomposition
Type of vector space in math
sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of orthogonal projection operators. The
Hilbert_space
Mapping from a Euclidean space to itself
reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents
Reflection_(mathematics)
In linear algebra, the complete orthogonal decomposition is a matrix decomposition. It is similar to the singular value decomposition, but typically somewhat
Complete orthogonal decomposition
Complete_orthogonal_decomposition
Parameterization of a rotation into a unit vector and angle
Plugging the three eigenvalues 1 and e±iθ and their associated three orthogonal axes in a Cartesian representation into Mercer's theorem is a convenient
Axis–angle_representation
Variant of draughts played in the Mediterranean and Middle East
Men move orthogonally forwards or sideways one square, capturing by means of a jump; they cannot move or capture backwards or diagonally. When a man
Turkish_draughts
Complex matrix whose conjugate transpose equals its inverse
Skew-Hermitian matrix Matrix decomposition Orthogonal group O(n) Special orthogonal group SO(n) Orthogonal matrix Semi-orthogonal matrix Quantum logic gate Special
Unitary_matrix
Solid with four equal triangular faces
associated fundamental domains). The regular tetrahedron has two special orthogonal projections, one centered on a vertex or equivalently on a face, and one
Regular_tetrahedron
Multi-dimensional generalization of triangle
this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix Q = diag ( Q 1 , Q 2 , … ,
Simplex
Chess tactic
and checks, to advance a queen, rook, or king along a diagonal via a series of stepped orthogonal moves. This article uses algebraic notation to describe
Staircase_maneuver
Matrix decomposition
decomposed as A = Q R , {\displaystyle A=QR,} where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning Q T = Q − 1 {\displaystyle Q^{\textsf
QR_decomposition
Periodicity computation method
functions in A are orthogonal (that is, not correlated, meaning the columns have zero pair-wise dot products), the matrix ATA is diagonal; when the columns
Least-squares spectral analysis
Least-squares_spectral_analysis
Strategy board game
(vertically and horizontally) into the diagonal European framework. Pieces capture both diagonally and orthogonally, recovering some of Alquerque's original
Checkers
Statistical method in psychology
cross-loadings) There are two main types of factor rotation: orthogonal and oblique rotation. Orthogonal rotations constrain factors to be perpendicular to each
Exploratory_factor_analysis
Matrix operation which flips a matrix over its diagonal
whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if A T = A − 1 . {\displaystyle \mathbf {A} ^{\text{T}}=\mathbf
Transpose
Curvilinear coordinate system
surfaces are not orthogonal, as in orthogonal coordinates. Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since
Skew_coordinates
Matrix B such that B² equals a given matrix A
&2\\4&7\end{bmatrix}}} . When A is symmetric, the diagonalizing matrix V can be made an orthogonal matrix by suitably choosing the eigenvectors (see spectral
Square_root_of_a_matrix
Polynomial sequence
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets
Hermite_polynomials
Group of unitary matrices
this J {\displaystyle J} is orthogonal; writing all the groups as matrix groups fixes a J {\displaystyle J} (which is orthogonal) and ensures compatibility)
Unitary_group
Statistical theorem in the analysis of variance
simultaneously diagonalized by an orthogonal matrix and that their non-zero eigenvalues are all equal to +1. Once that's shown, take this orthogonal transform
Cochran's_theorem
Three-dimensional chess variant by Gary Gygax
capture any number of steps orthogonally or diagonally (like a chess queen). On levels 1 and 3: can move one step orthogonally (like a wazir). On any level:
Dragonchess
Q-analog in combinatorial mathematics
ISBN 9780521782012. Ismail, Mourad E. H.; Zhang, Ruiming (1994). "Diagonalization of certain integral operators". Advances in Mathematics. 108 (1): 1–33
Q-exponential
Triangle containing a 90-degree angle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular
Right_triangle
Set of chess variants played on a board with hexagonal cells
generally increased mobility for pieces (that cannot move diagonally) compared to a standard orthogonal chessboard. For example, a rook on a hexboard usually
Hexagonal_chess
Type of matrix representation
a positive semi-definite Hermitian matrix ( U {\displaystyle U} is an orthogonal matrix, and P {\displaystyle P} is a positive semi-definite symmetric
Polar_decomposition
is orthonormal when they are all unit vectors and are pairwise orthogonal. orthogonal matrix A real square matrix with rows (or columns) that form an
Glossary_of_linear_algebra
Geometric symmetry operation
the diagonal, and, together with the identity, is the center of the orthogonal group O ( n ) {\displaystyle O(n)} . It is a product of n orthogonal reflections
Point_reflection
Matrix representing a Euclidean rotation
1. The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO(n), one example
Rotation_matrix
Physics software package
method for Coulombic energies. Along with plane waves and iterative diagonalization methods (via conjugate gradient or blocked Davidson algorithms), pseudopotentials
CASTEP
Movement with a fixed point is rotation
any orthogonal matrix R corresponding to a proper rotation is equivalent to a rotation over an angle φ around an axis n. The trace (sum of diagonal elements)
Euler's_rotation_theorem
Mathematics of number-placement puzzle
symmetry, which also includes a symmetry on both orthogonal axis, 180° rotational symmetry, and diagonal symmetry) is known to exist, but it is not known
Mathematics_of_Sudoku
Conic sections with the same foci
confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally (at right angles). Parabolas have only one focus, so, by convention,
Confocal_conic_sections
Model for approximating non-linear effects, similar to a Taylor series
identification orthogonalization, Volterra series must be rearranged in terms of orthogonal non-homogeneous G operators (Wiener series): y ( n ) = ∑ p H p x ( n )
Volterra_series
Type of mathematical space
replaced by the orthogonal group O(n), and Tn by the diagonal orthogonal matrices (which have diagonal entries ±1). The partial flag variety F ( d 1 , d
Generalized_flag_variety
Line joining midpoints of a complete quadrilateral's 3 diagonals
PNM-\angle FNM,\\&=\angle PNF=\angle BEF.\end{aligned}}} Let G and H be the orthogonal projections of the point F on the lines AB and CD respectively. The quadrilaterals
Newton–Gauss_line
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ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
ORTHOGONAL DIAGONALIZATION
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