Central object in linear algebra; mapping vectors to vectors
In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then
for some matrix , called the transformation matrix of .[citation needed] Note that has rows and columns, whereas the transformation is from to . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.[1][2]
^Rafael Artzy (1965) Linear Geometry
^J. W. P. Hirschfeld (1979) Projective Geometry of Finite Fields, Clarendon Press
and 27 Related for: Transformation matrix information
} for some m × n {\displaystyle m\times n} matrix A {\displaystyle A} , called the transformationmatrix 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...
&\lbrack b_{n}\rbrack _{C}\end{bmatrix}}} This matrix is referred to as the basis transformationmatrix from B to C. It can be regarded as an automorphism...
dealt further in this article. Writing the general matrixtransformation of coordinates as the matrix equation [ x ′ 0 x ′ 1 x ′ 2 x ′ 3 ] = [ Λ 0 0 Λ 0...
Similarity transformation may refer to: Similarity (geometry), for shape-preserving transformationsMatrix similarity, for matrixtransformations 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 =...
shear transformation, transvection, or just shearing. The transformations can be applied with a shear matrix or transvection, an elementary matrix that...
example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous...
a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformationmatrix, which can be applied to a signal through matrix multiplication...
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 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 transformationmatrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [...
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...
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...
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...
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...
Though matrix representations are not strictly necessary for Galilean transformation, they provide the means for direct comparison to transformation methods...
Bogoliubov transformations are linear recombination of operators, it is more convenient and insightful to write them in terms of matrixtransformations. If a...
contravariant transformations, with one transformation law for each index. If the transformationmatrix of an index is the inverse matrix of the basis...
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...
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...
whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into...
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 transformationmatrix from the frame of link i {\displaystyle i} to link i − 1 {\displaystyle...
Affine transformation, in geometry Linear transformation between modules in linear algebra. Also called a linear map. Transformationmatrix which represent...
its transformationmatrix is first computed to transform the points, followed by multiplying the propeller model matrix to the aircraft's matrix for the...
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