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 )...

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...

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

&\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...

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...

elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f. For example, the 2×2 matrix A = [ a c b d...

applied mathematics, a DFT matrix is a square matrix as an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied...

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 = [...

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

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...

} 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...

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...

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

In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay...

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

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

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

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...

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

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...

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

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

special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a diagonal matrix whose entries on...

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...

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...

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

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