} for some m × n {\displaystyle m\times n} matrix A {\displaystyle A} , called the transformation matrix of T {\displaystyle T} .[citation needed] Note...

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...

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

conjugate transpose v * {\textstyle v^{\textsf {*}}} . The matrix constructed from this transformation can be expressed in terms of an outer product as: P =...

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

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...

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

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...

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

} 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 = [...

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...

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...

Affine transformation, in geometry Linear transformation between modules in linear algebra. Also called a linear map. Transformation matrix which represent...

constant or multiplier of the transformation. The transformation is said to be elliptic if it can be represented by a matrix H {\displaystyle {\mathfrak...

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...

component). The power-invariant, right-handed, uniformly-scaled Clarke transformation matrix is K C = 2 3 ⋅ [ 1 − 1 2 − 1 2 0 3 2 − 3 2 1 2 1 2 1 2 ] {\displaystyle...

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...

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

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

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...

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

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...

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...

1 T i ( θ i ) {\displaystyle {}^{i-1}T_{i}(\theta _{i})} is the transformation matrix from the frame of link i {\displaystyle i} to link i − 1 {\displaystyle...

Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods...

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...