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| In [[linear algebra]], a '''circulant matrix''' is a special kind of [[Toeplitz matrix]] where each [[row vector]] is rotated one element to the right relative to the preceding row vector. In [[numerical analysis]], circulant matrices are important because they are diagonalized by a [[discrete Fourier transform]], and hence [[linear equation]]s that contain them may be quickly solved using a [[fast Fourier transform]].<ref>[[Philip J. Davis|Davis, Philip J.]], Circulant Matrices, Wiley, New York, 1970 ISBN 0471057711</ref> They can be [[#Analytic interpretation|interpreted analytically]] as the [[integral kernel]] of a [[convolution operator]] on the [[cyclic group]] <math>\mathbf{Z}/n\mathbf{Z}.</math>
| | You may find easy ways to accelerate computer by creating the many from the built inside tools in a Windows also as downloading the Service Pack updates-speed up the PC and fix error. Simply follow a few regulations to swiftly make the computer quick than ever.<br><br>Google Chrome crashes on Windows 7 by the corrupted cache contents and problems with all the stored browsing data. Delete the browsing information plus well-defined the contents of the cache to resolve this problem.<br><br>Windows is especially dumb. It only knows how to follow commands plus instructions, meaning which whenever you install a program, which program has to tell Windows precisely what to do. This really is done by storing an "training file" inside the registry of your program. All the computer programs place these "manuals" into the registry, allowing the computer to run a broad range of programs. When you load up 1 of those programs, Windows merely looks up the system file in the registry, and carries out its instructions.<br><br>Chrome allows customizing itself by applying range of themes accessible online. If you had newly applied a theme which no longer functions properly, it results inside Chrome crash on Windows 7. It is suggested to set the authentic theme.<br><br>Another way when arresting the 1328 error is to clean out your PC's registry. The registry is important as it's where settings plus files utilized by Windows for running are stored. As it is actually frequently utilized, breakdowns plus situations of files getting corrupted are not unusual. Also considering of the method it really is configured, the "registry" gets saved inside the incorrect fashion frequently, which makes the system run slow, ultimately causing a PC to suffer from a series of mistakes. The best way one may use inside cleaning out registries is to use a reliable [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014] system. A registry cleaner could find out and repair corrupted registry files and settings allowing one's computer to run usually again.<br><br>Files with all the DOC extension are moreover susceptible to viruses, however, this is solved by good antivirus programs. Another issue is that .doc files might be corrupted, unreadable or damaged due to spyware, adware, plus malware. These situations will avoid consumers from correctly opening DOC files. This is when effective registry cleaners become valuable.<br><br>Why why this really is important, is considering most 'dumb' registry products actually delete these files without even knowing. They simply browse through your registry and try and find the most problems possible. They then delete any files they see fit, plus considering they are 'dumb', they don't actually care. This signifies that when they delete some of these vital system files, they are actually going to cause a LOT more damage than advantageous.<br><br>There are many firms that offer the service of troubleshooting a PC every time we call them, all you need to do is sign up with them and for a small fee, you could have the machine usually functioning effectively plus serve you better. |
| In [[cryptography]], a circulant matrix is used in the [[Rijndael mix columns|MixColumns]] step of the [[Advanced Encryption Standard]].
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| ==Definition==
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| An <math>n\times n</math> circulant matrix <math>\ C</math> takes the form
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| :<math>
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| C=
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| \begin{bmatrix}
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| c_0 & c_{n-1} & \dots & c_{2} & c_{1} \\
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| c_{1} & c_0 & c_{n-1} & & c_{2} \\
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| \vdots & c_{1}& c_0 & \ddots & \vdots \\
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| c_{n-2} & & \ddots & \ddots & c_{n-1} \\
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| c_{n-1} & c_{n-2} & \dots & c_{1} & c_0 \\
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| \end{bmatrix}.
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| </math>
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| A circulant matrix is fully specified by one vector, <math>\ c</math>, which appears as the first column of <math>\ C</math>. The remaining columns of <math>\ C</math> are each [[cyclic permutation]]s of the vector <math>\ c</math> with offset equal to the column index. The last row of <math>\ C</math> is the vector <math>\ c</math> in reverse order, and the remaining rows are each [[cyclic permutation]]s of the last row. Note that different sources define the circulant matrix in different ways, for example with the coefficients corresponding to the first row rather than the first column of the matrix, or with a different direction of shift.
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| The polynomial <math> f(x) = c_0 + c_1 x + \dots + c_{n-1} x^{n-1} </math> is called the ''associated polynomial'' of matrix <math> C </math>.
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| == Properties ==
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| === Eigenvectors and eigenvalues ===
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| The eigenvectors of a circulant matrix are given by
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| :<math>v_j=(1,~ \omega_j,~ \omega_j^2,~ \ldots,~ \omega_j^{n-1})^T,\quad j=0, 1,\ldots, n-1,</math>
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| where <math>\omega_j=\exp \left(\tfrac{2\pi i j}{n}\right)</math> are the ''n''-th [[roots of unity]] and <math>i=\sqrt{-1}</math> is the [[imaginary unit]].
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| The corresponding eigenvalues are then given by
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| :<math>\lambda_j = c_0+c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \ldots + c_{1} \omega_j^{n-1}, \qquad j=0\ldots n-1.</math>
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| === Determinant ===
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| As a consequence of the explicit formula for the eigenvalues above,
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| the [[determinant]] of circulant matrix can be computed as:
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| :<math>
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| \mathrm{det}(C)
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| = \prod_{j=0}^{n-1} (c_0 + c_{n-1} \omega_j + c_{n-2} \omega_j^2 + \dots + c_1\omega_j^{n-1}).</math>
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| Since taking transpose does not change the eigenvalues of a matrix, an equivalent formulation is
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| :<math>
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| \mathrm{det}(C)=\prod_{j=0}^{n-1} (c_0 + c_1 \omega_j + c_2 \omega_j^2 + \dots + c_{n-1}\omega_j^{n-1}) = \prod_{j=0}^{n-1} f(\omega_j).
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| </math> | |
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| === Rank ===
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| The [[Rank (linear algebra)|rank]] of circulant matrix <math> C </math> is equal to <math> n - d </math>, where <math> d </math> is the [[degree of a polynomial|degree]] of <math> \gcd( f(x), x^n - 1) </math>.<ref>{{cite journal |author=A. W. Ingleton |title=The Rank of Circulant Matrices |journal=J. London Math. Soc. |year=1956 |volume=s1-31 |issue=4 |pages=445-460 |doi=10.1112/jlms/s1-31.4.445}}</ref>
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| === Other properties ===
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| * We have
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| ::<math> C=c_0I+c_{1}P+c_{2}P^2+\ldots+c_{n-1}P^{n-1}=f(P).</math>
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| :where ''P'' is the 'cyclic permutation' matrix, a specific [[permutation matrix]] given by
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| ::<math>P=
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| \begin{bmatrix}
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| 0&0&\ldots&0&1\\
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| 1&0&\ldots&0&0\\
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| 0&\ddots&\ddots&\vdots&\vdots\\
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| \vdots&\ddots&\ddots&0&0\\
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| 0&\ldots&0&1&0
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| \end{bmatrix}.</math>
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| * The [[Set (mathematics)|set]] of <math>n\times n</math> circulant matrices forms an ''n''-[[dimensional]] [[vector space]]; this can be interpreted as the space of functions on the [[cyclic group]] of order ''n'', <math>\mathbf{Z}/n\mathbf{Z},</math> or equivalently the [[group ring]].
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| * Circulant matrices form a [[commutative algebra]], since for any two given circulant matrices <math>\ A</math> and <math>\ B</math>, the sum <math>\ A + B</math> is circulant, the product <math>\ AB</math> is circulant, and <math>\ AB = BA</math>.
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| * The [[eigenvectors]] of a circulant matrix of a given size are the columns of the [[Discrete Fourier transform#The unitary DFT|unitary discrete Fourier transform]] matrix of the same size. The latter matrix is defined by
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| ::<math> U_n = \frac{1}{\sqrt{n}} F_n, \quad\text{where}\quad F_n = (f_{jk}) \quad\text{with}\quad f_{jk} = \mathrm{e}^{-2jk\pi\mathrm{i}/n}, \quad\text{for}\quad 0\leq j,k<n.</math>
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| :Thus, the matrix <math>U_n</math> [[diagonalizable matrix|diagonalizes]] ''C''. In fact, we have
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| ::<math> C = U_n^{*} \operatorname{diag}(F_n c) U_n = \frac{1}{n} F_n^{*} \operatorname{diag}(F_n c) F_n, </math>
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| :where <math>c\!\,</math> is the first column of <math>C\,\!</math>. Thus, the eigenvalues of <math>C</math> are given by the product <math>\ F_n c</math>. This product can be readily calculated by a [[Fast Fourier transform]].<ref>{{Citation | last1=Golub | first1=Gene H. | author1-link=Gene H. Golub | last2=Van Loan | first2=Charles F. | author2-link=Charles F. Van Loan | title=Matrix Computations | chapter=§4.7.7 Circulant Systems | publisher=Johns Hopkins | edition=3rd | isbn=978-0-8018-5414-9 | year=1996}}</ref> | |
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| ==Analytic interpretation==
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| Circulant matrices can be interpreted geometrically, which explains the connection with the discrete Fourier transform.
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| Consider vectors in <math>\mathbf{R}^n</math> as functions on the integers with period ''n,'' (i.e., as periodic bi-infinite sequences: <math>\dots,a_0,a_1,\dots,a_{n-1},a_0,a_1,\dots</math>) or equivalently, as functions on the [[cyclic group]] of order ''n,'' (<math>C_n</math> or <math>\mathbf{Z}/n\mathbf{Z}</math>) geometrically, on (the vertices of) the regular ''n''-gon: this is a discrete analog to periodic functions on the real line or circle.
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| Then, from the perspective of [[operator theory]], a circulant matrix is the kernel of a discrete [[integral transform]], namely the [[convolution operator]] for the function <math>(c_0,c_1,\dots,c_{n-1});</math> this is a discrete [[circular convolution]]. The formula for the convolution of the functions <math>(b_i) := (c_i) * (a_i)</math> is
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| :<math>b_k = \sum_{i=0}^{n-1} a_i c_{k-i}</math> (recall that the sequences are periodic)
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| which is the product of the vector of <math>a_i</math> by the circulant matrix.
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| The discrete Fourier transform then converts convolution into multiplication, which in the matrix setting corresponds to diagonalization.
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| == Applications ==
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| ===In linear equations===
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| Given a matrix equation
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| :<math>\ \mathbf{C} \mathbf{x} = \mathbf{b},</math>
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| where <math>\ C</math> is a circulant square matrix of size <math>\ n</math> we can write the equation as the [[circular convolution]]
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| :<math>\ \mathbf{c} \star \mathbf{x} = \mathbf{b},</math>
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| where <math>\ c</math> is the first column of <math>\ C</math>, and the vectors <math>\ c</math>, <math>\ x</math> and <math>\ b</math> are cyclically extended in each direction. Using the results of the [[discrete Fourier transform#Circular convolution theorem and cross-correlation theorem|circular convolution theorem]], we can use the [[discrete Fourier transform]] to transform the cyclic convolution into component-wise multiplication
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| :<math>\ \mathcal{F}_{n}(\mathbf{c} \star \mathbf{x}) = \mathcal{F}_{n}(\mathbf{c}) \mathcal{F}_{n}(\mathbf{x}) = \mathcal{F}_{n}(\mathbf{b})</math>
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| so that
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| :<math>\ \mathbf{x} = \mathcal{F}_{n}^{-1}
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| \left [
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| \left (
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| \frac{(\mathcal{F}_n(\mathbf{b}))_{\nu}}
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| {(\mathcal{F}_n(\mathbf{c}))_{\nu}}
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| \right )_{\nu \in \mathbf{Z}}
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| \right ]^T.
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| </math>
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| This algorithm is much faster than the standard [[Gaussian elimination]], especially if a [[fast Fourier transform]] is used.
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| === In graph theory ===
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| In [[graph theory]], a [[Graph (mathematics)|graph]] or [[Directed graph|digraph]] whose [[adjacency matrix]] is circulant is called a [[circulant graph]] (or digraph). Equivalently, a graph is circulant if its [[automorphism group]] contains a full-length cycle. The [[Möbius ladder]]s are examples of circulant graphs, as are the [[Paley graph]]s for fields of prime order.
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| ==References==
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| {{reflist}}
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| ==External links==
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| * R. M. Gray, [http://www-ee.stanford.edu/~gray/toeplitz.pdf Toeplitz and Circulant Matrices: A Review]
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| * {{MathWorld|urlname=CirculantMatrix|Circulant Matrix}}
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| {{Numerical linear algebra}}
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| [[Category:Numerical linear algebra]]
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| [[Category:Matrices]]
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| [[Category:Latin squares]]
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| [[Category:Determinants]]
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You may find easy ways to accelerate computer by creating the many from the built inside tools in a Windows also as downloading the Service Pack updates-speed up the PC and fix error. Simply follow a few regulations to swiftly make the computer quick than ever.
Google Chrome crashes on Windows 7 by the corrupted cache contents and problems with all the stored browsing data. Delete the browsing information plus well-defined the contents of the cache to resolve this problem.
Windows is especially dumb. It only knows how to follow commands plus instructions, meaning which whenever you install a program, which program has to tell Windows precisely what to do. This really is done by storing an "training file" inside the registry of your program. All the computer programs place these "manuals" into the registry, allowing the computer to run a broad range of programs. When you load up 1 of those programs, Windows merely looks up the system file in the registry, and carries out its instructions.
Chrome allows customizing itself by applying range of themes accessible online. If you had newly applied a theme which no longer functions properly, it results inside Chrome crash on Windows 7. It is suggested to set the authentic theme.
Another way when arresting the 1328 error is to clean out your PC's registry. The registry is important as it's where settings plus files utilized by Windows for running are stored. As it is actually frequently utilized, breakdowns plus situations of files getting corrupted are not unusual. Also considering of the method it really is configured, the "registry" gets saved inside the incorrect fashion frequently, which makes the system run slow, ultimately causing a PC to suffer from a series of mistakes. The best way one may use inside cleaning out registries is to use a reliable tuneup utilities 2014 system. A registry cleaner could find out and repair corrupted registry files and settings allowing one's computer to run usually again.
Files with all the DOC extension are moreover susceptible to viruses, however, this is solved by good antivirus programs. Another issue is that .doc files might be corrupted, unreadable or damaged due to spyware, adware, plus malware. These situations will avoid consumers from correctly opening DOC files. This is when effective registry cleaners become valuable.
Why why this really is important, is considering most 'dumb' registry products actually delete these files without even knowing. They simply browse through your registry and try and find the most problems possible. They then delete any files they see fit, plus considering they are 'dumb', they don't actually care. This signifies that when they delete some of these vital system files, they are actually going to cause a LOT more damage than advantageous.
There are many firms that offer the service of troubleshooting a PC every time we call them, all you need to do is sign up with them and for a small fee, you could have the machine usually functioning effectively plus serve you better.