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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:

${\displaystyle \mathrm{vec}(A)=[a_{1,1},\dots ,a_{m,1},a_{1,2},\dots ,a_{m,2},\dots ,a_{1,n},\dots ,a_{m,n}{]}^{\mathrm{T}}}$

Here, ${\displaystyle a_{i,j}}$ represents ${\displaystyle A(i,j)}$ and the superscript ${\displaystyle {\mathrm{T}}^{}}$ denotes the transpose. Vectorization expresses, through coordinates, the isomorphism${\displaystyle {\mathbf{R}}^{m\times n}:={\mathbf{R}}^m\otimes {\mathbf{R}}^n\cong {\mathbf{R}}^{mn}}$ between these (i.e., of matrices and vectors) as vector spaces.

For example, for the 2×2 matrix , the vectorization is ${\displaystyle \mathrm{vec}(A)=\left[\begin{array}{c}a\\ c\\ b\\ d\end{array}\right]}$.

The connection between the vectorization of A and the vectorization of its transpose is given by the commutation matrix.

Compatibility with Kronecker products[edit]

The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular,

${\displaystyle \mathrm{vec}(ABC)=(C^{\mathrm{T}}\otimes A)\mathrm{vec}(B)}$

for matrices A, B, and C of dimensions k×l, l×m, and m×n.^[1] For example, if ${\displaystyle {\mathrm{ad}}_A(X)=AX-XA}$ (the adjoint endomorphism of the Lie algebragl(n, C) of all n×n matrices with complex entries), then ${\displaystyle \mathrm{vec}({\mathrm{ad}}_A(X))=(I_n\otimes A-A^{\mathrm{T}}\otimes I_n)\text{vec}(X)}$, where ${\displaystyle I_n}$ is the n×nidentity matrix.

There are two other useful formulations:

${\displaystyle \mathrm{vec}(ABC)=(I_n\otimes AB)\mathrm{vec}(C)=(C^{\mathrm{T}}{B}^{\mathrm{T}}\otimes I_k)\mathrm{vec}(A)}$

${\displaystyle \mathrm{vec}(AB)=(I_m\otimes A)\mathrm{vec}(B)=(B^{\mathrm{T}}\otimes I_k)\mathrm{vec}(A)}$

More generally, it has been shown that vectorization is a self-adjunction in the monoidal closed structure of any category of matrices.^[1]

Compatibility with Hadamard products[edit]

Vectorization is an algebra homomorphism from the space of n × n matrices with the Hadamard (entrywise) product to Cⁿ² with its Hadamard product:

${\displaystyle \mathrm{vec}(A\circ B)=\mathrm{vec}(A)\circ \mathrm{vec}(B).}$

Compatibility with inner products[edit]

Vectorization is a unitary transformation from the space of n×n matrices with the Frobenius (or Hilbert–Schmidt) inner product to Cⁿ²:

${\displaystyle \mathrm{tr}(A^{\mathrm{\top }}B)=\mathrm{vec}(A{)}^{\mathrm{\top }}\mathrm{vec}(B)=\mathrm{vec}(B{)}^{\mathrm{\top }}\mathrm{vec}(A).}$

where the superscript ^T denotes the conjugate transpose.

Vectorization as a linear sum[edit]

The matrix vectorization operation can be written in terms of a linear sum. Let X be an m × n matrix that we want to vectorize, and let e_i be the i-th canonical basis vector for the n-dimensional space, that is ${\mathbf{e}}_i={\left[0,...,0,1,0,...,0\right]}^{\mathrm{T}}$. Let B_i be a (mn) × m block matrix defined as follows:

${\displaystyle {\mathbf{B}}_i=\left[\begin{array}{c}\mathbf{0}\\ ⋮\\ \mathbf{0}\\ {\mathbf{I}}_m\\ \mathbf{0}\\ ⋮\\ \mathbf{0}\end{array}\right]={\mathbf{e}}_i\otimes {\mathbf{I}}_m}$

B_i consists of n block matrices of size m × m, stacked column-wise, and all these matrices are all-zero except for the i-th one, which is a m × m identity matrix I_m.

Then the vectorized version of X can be expressed as follows:

${\displaystyle \mathrm{vec}(\mathbf{X})=\sum _{i=1}^n{\mathbf{B}}_i\mathbf{X}{\mathbf{e}}_i}$

Multiplication of X by e_i extracts the i-th column, while multiplication by B_i puts it into the desired position in the final vector.

Alternatively, the linear sum can be expressed using the Kronecker product:

${\displaystyle \mathrm{vec}(\mathbf{X})=\sum _{i=1}^n{\mathbf{e}}_i\otimes \mathbf{X}{\mathbf{e}}_i}$

Half-vectorization[edit]

For a symmetric matrixA, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the vectorization. The half-vectorization, vech(A), of a symmetric n × n matrix A is the n(n + 1)/2 × 1 column vector obtained by vectorizing only the lower triangular part of A:

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${\displaystyle \mathrm{vech}(A)=[A_{1,1},\dots ,A_{n,1},A_{2,2},\dots ,A_{n,2},\dots ,A_{n-1,n-1},A_{n,n-1},A_{n,n}{]}^{\mathrm{T}}.}$

For example, for the 2×2 matrix , the half-vectorization is ${\displaystyle \mathrm{vech}(A)=\left[\begin{array}{c}a\\ b\\ d\end{array}\right]}$.

There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice versa called, respectively, the duplication matrix and the elimination matrix.

Programming language[edit]

Programming languages that implement matrices may have easy means for vectorization.In Matlab/GNU Octave a matrix A can be vectorized by A(:).GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well.In PythonNumPy arrays implement the flatten method^[1], while in R the desired effect can be achieved via the c() or as.vector() functions. In R, function vec() of package 'ks' allows vectorization and function vech() implemented in both packages 'ks' and 'sn' allows half-vectorization.^[2][3][4]

Notes[edit]

1.^{^^} The identity for row-major vectorization is ${\displaystyle \mathrm{vec}(ABC)=(A\otimes C^{\mathrm{T}})\mathrm{vec}(B)}$.

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See also[edit]

References[edit]

1. ^Macedo, H. D.; Oliveira, J. N. (2013). 'Typing Linear Algebra: A Biproduct-oriented Approach'. Science of Computer Programming. 78 (11): 2160–2191. arXiv:1312.4818. doi:10.1016/j.scico.2012.07.012. S2CID9846072.
2. ^Duong, Tarn (2018). 'ks: Kernel Smoothing'. R package version 1.11.0.
3. ^Azzalini, Adelchi (2017). 'The R package 'sn': The Skew-Normal and Related Distributions such as the Skew-t'. R package version 1.5.1.
4. ^Vinod, Hrishikesh D. (2011). 'Simultaneous Reduction and Vec Stacking'. Hands-on Matrix Algebra Using R: Active and Motivated Learning with Applications. Singapore: World Scientific. pp. 233–248. ISBN978-981-4313-69-8 – via Google Books.

• Jan R. Magnus and Heinz Neudecker (1999), Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd Ed., Wiley. ISBN0-471-98633-X.

What is a vector and how is it different from other image types?

Vector images are path-based mathematical equations which allow unlimited resizing without degradation of image quality, or to put it simply, the image remains smooth without pixelation or jagged edges.

> Find out what makes up a vector file and how to create one.

> Create a Vector Portrait from a Photograph

These differ from Raster or Bitmap images which are made up of pixels and are resolution-dependent. This means enlarging the image will increase the size of the pixels making the image look grainy or blotchy with jagged lines and edges.

Zapgrab free download for mac. There are good reason for having these two different styles of image file as some images aren’t practical as a vector. Photographs, as an example, don’t work well as vectors due to their complexity. The different shades, gradients, colors and irregular shapes make it difficult or impractical to be saved as a vector.

Vector images are best suited for images or designs with smooth lines, curves, and shapes with solid colors. Typically this includes logos, graphic art, line drawings and illustrations.

Converting an Image to a Vector

The conversion of bitmap or raster graphics to vector art is called ‘Vectorizing’, and can be done a number of ways. There are programs that can be used to convert bitmap images into vector images automatically such as Gravit Designer, Vector Magic, and Vectorizer.io

You can automatically convert your image to a vector in Gravit Designer for free by importing your image and selecting Modify > Path > Vectorize Image from the App menu.

The other way to convert is through a method called tracing. Not unlike using tracing paper to create an illustration from a photograph, you would import your image into a vector design program (such as Gravit Designer or Adobe Illustrator) and use paths to trace the lines and elements you wish to reproduce.

Examples of Bitmap or Raster filetypes:

• .jpeg and .jpg
• .png
• .bmp
• .gif

Examples of Vector filetypes:

• .svg
• .eps
• .pdf
• .ai

Gravit Designer is a Free vector design app for all platforms- sign up and start designing today!