About: Vectorization (mathematics)

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In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another: Here, represents and the superscript denotes the transpose. Vectorization expresses, through coordinates, the isomorphism between these (i.e., of matrices and vectors) as vector spaces. For example, for the 2×2 matrix , the vectorization is .

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rdfs:label	Vec作用素 (ja) 벡터화 (ko) Vectorization (mathematics) (en)
rdfs:comment	vec作用素（英語: vec operator）とは m × n 行列 A の要素を mn 次元列ベクトルの形に配置し直す作用素である。vec作用素は行列の微分を行うのに便利なことがある。 m × n 行列 A を m 次元列ベクトルを用いてと書けるとき、として定義される。 (ja) 수학에서, 특히 선형대수학과 행렬 이론에서 행렬의 벡터화(Vector化, 영어:Vectorization)는 행렬을 세로 벡터로 바꾸는 선형변환의 하나이다. m×n행렬 A의 선형화는 vec(A)로 표기하며, 행렬 A의 열을 다음 열 위에 쌓아가며 얻을 수 있다. 는 행렬 의 성분을 나타내며, 는 전치행렬을 나타낸다. 벡터화는 (행렬과 벡터의)벡터 공간 사이의 동형 사상 을 나타낸다. 예를 들어, 2×2 행렬 = 를 벡터화하면가 된다. (ko) In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another: Here, represents and the superscript denotes the transpose. Vectorization expresses, through coordinates, the isomorphism between these (i.e., of matrices and vectors) as vector spaces. For example, for the 2×2 matrix , the vectorization is . (en)
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dcterms:subject	Matrices Linear algebra
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Link from a Wikipage to another Wikipage	Python (programming language) Algebra homomorphism Julia (programming language) Packed storage matrix Column vector Complex number Conjugate transpose Mathematics Matlab Matrix (mathematics) Matrix multiplication Matrix norm Symmetric matrix GNU Octave Lie algebra Linear algebra Commutation matrix Identity matrix Main diagonal Matricization Adjoint functors Matrices Transpose Hadamard product (matrices) Linear algebra NumPy Hilbert–Schmidt operator Isomorphism Duplication matrix Unitary transformation Inner product Kronecker product Voigt notation Linear transformation Row-major order R programming language Elimination matrix Lower triangular matrix Adjoint endomorphism
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has abstract	In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another: Here, represents and the superscript denotes the transpose. Vectorization expresses, through coordinates, the isomorphism between these (i.e., of matrices and vectors) as vector spaces. For example, for the 2×2 matrix , the vectorization is . The connection between the vectorization of A and the vectorization of its transpose is given by the commutation matrix. (en) vec作用素（英語: vec operator）とは m × n 行列 A の要素を mn 次元列ベクトルの形に配置し直す作用素である。vec作用素は行列の微分を行うのに便利なことがある。 m × n 行列 A を m 次元列ベクトルを用いてと書けるとき、として定義される。 (ja) 수학에서, 특히 선형대수학과 행렬 이론에서 행렬의 벡터화(Vector化, 영어:Vectorization)는 행렬을 세로 벡터로 바꾸는 선형변환의 하나이다. m×n행렬 A의 선형화는 vec(A)로 표기하며, 행렬 A의 열을 다음 열 위에 쌓아가며 얻을 수 있다. 는 행렬 의 성분을 나타내며, 는 전치행렬을 나타낸다. 벡터화는 (행렬과 벡터의)벡터 공간 사이의 동형 사상 을 나타낸다. 예를 들어, 2×2 행렬 = 를 벡터화하면가 된다. (ko)
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is Link from a Wikipage to another Wikipage of	Bayesian multivariate linear regression Vector (mathematics and physics) Duplication and elimination matrices List of named matrices Elliptical distribution Low-rank approximation Superoperator Quantum Fisher information Commutation matrix Frobenius inner product Machine learning in bioinformatics Matrix normal distribution Trace (linear algebra) A Dynamical Theory of the Electromagnetic Field Exponential family Fang Kaitai Stochastic gradient descent Inverse-Wishart distribution Spatial correlation Transportation theory (mathematics) Mode-k flattening Bures metric Kronecker product

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