there exists an m × n {\displaystyle m\times n} matrix A {\displaystyle A} , called the transformation matrix of T {\displaystyle T} , such that: T ( x )...

an affine transformation is invertible, the square matrix A {\displaystyle A} appearing in its matrix representation is invertible. The matrix representation...

dealt further in this article. Writing the general matrix transformation of coordinates as the matrix equation [ x ′ 0 x ′ 1 x ′ 2 x ′ 3 ] = [ Λ 0 0 Λ 0...

&\lbrack b_{n}\rbrack _{C}\end{bmatrix}}} This matrix is referred to as the basis transformation matrix from B to C. It can be regarded as an automorphism...

square matrix are the roots of a polynomial determinant. In geometry, matrices are widely used for specifying and representing geometric transformations (for...

transformation (named after Robert H. Park) is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix...

transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing...

shear transformation, transvection, or just shearing. The transformations can be applied with a shear matrix or transvection, an elementary matrix that...

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

a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication...

} A linear transformation L can be represented by a matrix, which means L : v → [L]v, where [L] is an n×n matrix. A linear transformation is a rigid transformation...

rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [...

Similarity transformation may refer to: Similarity (geometry), for shape-preserving transformations Matrix similarity, for matrix transformations of the form...

whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into...

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

example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous...

numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors...

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices...

the form of an n by n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication A v = λ...

be defined by a nonsingular (n+1) × (n+1) matrix [ai,j], called the matrix of the homography. This matrix is defined up to the multiplication by a nonzero...

being the change-of-basis matrix. A transformation A ↦ P−1AP is called a similarity transformation or conjugation of the matrix A. In the general linear...

Such a matrix is also called a Frobenius matrix, a Gauss matrix, or a Gauss transformation matrix. A block triangular matrix is a block matrix (partitioned...

contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis...

Transformation geometry Transformation semigroup Transformation group Transformation matrix "Self-Map -- from Wolfram MathWorld". Retrieved March 4, 2024. Olexandr...

its transformation matrix is first computed to transform the points, followed by multiplying the propeller model matrix by the aircraft's matrix for the...

.} An n × n matrix M can represent a linear map and act on row and column vectors as the linear map's transformation matrix. For a row vector v...

Bogoliubov transformations are linear recombination of operators, it is more convenient and insightful to write them in terms of matrix transformations. If a...

vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial...

the Hunt–Pointer–Estevez transformation matrix (MHPE) for conversion from CIE XYZ to LMS. This is the transformation matrix which was originally used...

the second transformation matrix in the above-mentioned general form is inverse of first transformation matrix. The transformation matrix should account...