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...pute and in particular much cheaper and easier to update when the original matrix is slightly altered.<ref name="golub-van-loan" /> ...trix <math>A</math> of [[matrix rank|rank]] <math>r</math>, the triangular matrix <math>T</math> can be chosen such that only its top-left <math>r\times r</m ...

...th stemming from the fact that the rows and columns come from the original matrix (rather than left and right singular vectors): ...composed matrix are essentially the same as their meanings in the original matrix. ...

...nd the other of which has a subset of columns consisting of the [[identity matrix]] and all its values are no greater than&nbsp;2 in absolute value. ...n </math> matrix of [[Rank (linear algebra)|rank]] <math> r </math>. The matrix <math> A </math> can be written as ...

...21768 |issn=1053-587X}}</ref> Many tensor decompositions generalize some [[matrix decomposition]]s.<ref>{{Cite journal|date=2013-05-01|title=General tensor d ...|last3=Günnemann |first3=Stephan |date=2017 |title=Introduction to Tensor Decompositions and their Applications in Machine Learning |class=stat.ML |eprint=1711.1078 ...

...uations. The algorithm works by using the [[Schur decomposition|real Schur decompositions]] of <math>A</math> and <math>B</math> to transform <math> AX - XB = C</mat ...th>A</math> are distinct from the eigenvalues of <math>B</math>. Then, the matrix equation <math> AX - XB = C</math> has a unique solution. The Bartels–Stewa ...

...ath>, non-negative integers <math>m,n</math>, and a [[Matrix (mathematics)|matrix]] <math>A\in\mathbb F^{m\times n}</math>, a '''rank decomposition''' or ''' ...hbf{a}_n\end{bmatrix}</math> is an <math display="inline">m\times n</math> matrix with <math display="inline">\mathbf{a}_j</math> as the <math display="inlin ...

...on into modules with [[local ring|local]] [[endomorphism ring]]s, then all decompositions into indecomposable modules are equivalent to each other; a special case of ...f and only if it is a direct sum (in fact a [[product ring|product]]) of [[matrix ring]]s over [[division ring]]s (this observation is known as the [[Artin–W ...

The corresponding matrix form is ...iscrete gradient operator, and <math>A_{\operatorname{grad}}</math> is the matrix representation of the discrete scalar Laplacian defined on <math>H_h(\opera ...

...rs]] of a [[generic matrix|generic]] ''d''-by-''n'' [[matrix (mathematics)|matrix]] (''x''_''ij''). ...or ''syzygies'' that exist between the minors of a generic ''n'' by ''d'' matrix. The projective variety defined by the [[Ideal (ring theory)|ideal]] ''I'' ...

...ility theory]], Spectral Factorization has been used to find determinantal matrix representations for bivariate stable polynomials and real zero polynomials. ...al|last=Ephremidze|first=Lasha|title=An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem|journal=Proceedings of the Royal Society of ...

...<math>n-1</math> cells of an <math>n\times n</math> [[Matrix (mathematics)|matrix]] are preassigned with no element repeated in any row or column then the re ...

...of linear equations|linear system]], one can convert it to a [[Triangular matrix|triangular system]] via [[Gaussian elimination]]. For the non-linear case, ...decompositions of algebraic varieties. Without using factorization, these decompositions have better properties that the ones produced by [[Wu's method|Wu's algorit ...

...te the [[determinant]] or the [[echelon form]] of a [[Matrix (mathematics)|matrix]] with [[integer]] entries using only integer arithmetic; any [[division (m ...al Bareiss algorithm is distinct from the Bareiss algorithm for [[Toeplitz matrix|Toeplitz matrices]]. ...

...algebra)|basis]], composing with each-other and transforming vectors by [[matrix multiplication]]. The [[Lie group]] of these transformations has been calle ...es |first1=G.S. |last2=Wylie |first2=D. |year=2015 |title=Clifford algebra decompositions of conformal orthogonal group elements |journal=Clifford Analysis, Clifford ...

...that commute with matrix <math>B</math> are called the '''commutant''' of matrix <math>B</math> (and vice versa).<ref>Godsil, Christopher and Meagher, Karen ...r words, if <math>A_1,\ldots,A_k</math> commute, there exists a similarity matrix <math>P</math> such that <math>P^{-1} A_i P</math> is upper triangular for ...

In mathematics, the '''Hamiltonian cycle polynomial''' of an ''n''×''n''-matrix is a polynomial in its entries, defined as ...iltonian cycle polynomial of a matrix received from its weighted adjacency matrix via subjecting its rows and columns to any permutation mapping ''i'' to ''1 ...

...ion operations in graphs. The theory of matroid minors leads to structural decompositions of matroids, and characterizations of matroid families by forbidden minors, ...is regular if it can be represented by a [[totally unimodular matrix]] (a matrix whose square submatrices all have determinants equal to 0, 1, or &minus;1). ...

{{Short description|A matrix canonical form}} ...sisting of two blocks each of which is a basic Weyr matrix. The basic Weyr matrix in the top-left corner has the structure (4,2,1) and the other one has the ...

Here, <math>\varrho</math> is the [[density matrix|state]] of the system and <math>H</math> is the [[Hamiltonian (quantum mech Let us consider the decomposition of the density matrix to pure components as ...

More exotic examples include the [[upper triangular matrix|upper triangular matrices]] over a [[division ring]] T_''n''&thin ...[[subring]] of M_''n''(''V'') for some ''n'', and consists of [[matrix (mathematics)|matrices]] with entries from ''V'' on and above the diagonal, ...