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I blanked a section labeled "Algorithm", which presented a naive block of code doing .. something. There have got to dozens if not hundreds of algorithms that can be applied to triangular matrixes. This is not the right place for a compendium of these. Readers can be referred to github for LAPACK or
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The outline has a heading for "Forward and back substitution" with a sub section for "Forward substitution" but no subsection for
Backward substitution. Additionally, an equation is only given for forward sub. Furthermore, the algorithm provided for back sub is dependent on the first part solving
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My textbook (Spence, Insel, and
Friedberg. "Elementary Linear Algebra: A Matrix Approach", 2nd edition) in chapter 2.6 "The LU Decomposition of a Matrix" gives a 3x4 matrix as an example of an upper triangular matrix. Is it common to refer to non-square "trapezoidal" matrices as "triangular" in the
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The statement about simultaneous triangulation is false without further assumption (like diagonability of one of the matrices) It is false that two commuting matrices have a common eigenvector, we can find a conter example using a direct sum of two nilpotent Jordan blocks of the same size for the
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There is a lot of good material in here, but it seems to be arranged in no particular order. The level of exposition oscillates at high speed between what is appropriate for grade school and what is appropriate for graduate school. I am going to try to straighten things out a bit. Please help!
21:
The article clearly states that products of upper triangular matrices are upper triangular, but it doesn't make the similar (and also true) claim about lower triangular matrices. Further, I only vaguely get the impression that the inverses of upper/lower triangular matrices remain upper/lower
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IF A is real, there exists a real orthogonal matrix V such that V^T A V = T is quasi-upper triangular. This means that T is block upper triangular with 1-by1 and 2-by-2 blocks on the diagonal. Its eigenvalues are the eigenvalues of the diagonal blocks. The 1-by-1 blocks correspond to real
253:{\displaystyle \mathbf {L} ^{-1}={\begin{bmatrix}1&&&&&0\\&\ddots &&&&\\&&1&&&\\&&-l_{i+1,i}&\ddots &&\\&&\vdots &&\ddots &\\0&&-l_{n,i}&&&1\\\end{bmatrix}},}
554:{\displaystyle {\begin{bmatrix}1&&\\2&1&\\3&4&1\\\end{bmatrix}}{\begin{bmatrix}1&&\\-2&1&\\-3&-4&1\\\end{bmatrix}}={\begin{bmatrix}1&&\\0&1&\\-8&0&1\\\end{bmatrix}}\neq \mathbf {I} }
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This would have the added benefit that the reduced row echelon form admits a generalized backward substitution algorithm, which handles the case of underdetermined systems, and returns all possible solutions (see discussion, row echelon form)
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1305:(and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an
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Contrary to what this article claims, an upper-triangular matrix does NOT necessarily need to be square. I welcome someone who is familiar enough with the upper/lower definitions to fix this error.
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Is the claim about common eigenvalue wrong or I'm misinterpreting it? As far I know, two commuting matrices share a common eigenvector, but not necessarily a common eigenvalue: the identity matrix
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You are quite correct: reading through the article, the math typesetting looks like a general form lower triangular that's been normalized. Not good. I've extended the typesetting for the matrix
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it can be added: If the inverse U−1 of an upper triangular matrix U exists, then it is upper triangular. If the inverse L−1 of an lower triangular matrix L exists, then it is lower triangular.
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886:“Square” is generally required, square matrices being generally more interesting. For non-square matrices one generally calls these “trapezoidal” matrices, which is mentioned in the article.
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In MATLAB and related programs I have seen references to 'quasi-upper-triangular' matrices, but I can't find a definition. Would someone please add a definition here? --
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s “i” row-wise; “j” column-wise or “k” outer products Where i, j, k are referring to outermost loop index For column-wise there is a method 1 and 2 discussed. For
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to show all the lower diagonal zeros, and have added a section heading "special forms" to separate the paragraph from the general section on triangular matricies.
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the article says Gauss matrix only have 1 non-zero column below the diagonal. probably you didn't see that. for those matrices the claim holds trivially.
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going by the artcle's terminology, the matrix in your e.g. is not a "Gauss matrix". article only claims that formula holds when a matrix is Gauss.
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Each entry on the main diagonal of L-1 is equal to the reciprocal of the corresponding entry on the main diagonal of L.
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I'd therefore argue that this chapter, though generally formulated for triangular matrices, would be a better fit for
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triangular. We should probably state these properties more directly, and perhaps clean up the article in general. --
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Furthermore, the algorithm isn't well defined for triangular matrices that don't have a staircase form, e.g.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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context of LU Decompositions or is this something that is otherwise rare other than this book?
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Ly = b. No algorithm or equations are given for back sub of a given upper diagonal matrix.
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i wanna know if a null matrix would be called an upper triangular or lower triangular.
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for “Nilpotent”. This is a bit heavy duty (Lie algebra notation), but is a standard.
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Applied
Numerical Linear Algebra, James W. Demmel, 1997, copyright SIAM, page 147.
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Is there a standard notation for the algebra/ring of upper triangular matrices?--
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Some algorithms can be added from: "Stability of
Methods for Matrix Inversion"
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I've worked a bit on the first half now. The second half is untouched.
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first matrix and with the second matrix that permutes these blocks.
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A Venn diagram of types of triangular matrices would be helpful in
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quite as simple as reversing signs. Consider this counter example:
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I've added a paragraph about triangular matricies preserving form.
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The algorithm for forward substitution as it is now, assumes that
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https://epubs.siam.org/doi/abs/10.1137/0119075?journalCode=smjmap
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share common eigenvectors, but their eigenvalues are different.
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i.e. the off-diagonal entries are replaced by their opposites."
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eigenvalues, and the 2-by-2 blocks to complex conjugate pairs.
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http://homepages.warwick.ac.uk/~ecsgaj/matrixAlgSlidesC.pdf
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https://www.statlect.com/matrix-algebra/triangular-matrix
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about simultaneous triangularisability is claimed that
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The claim is still false. Look at the counter example.
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1187:: commuting matrices form a commutative algebra
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1162:Mid-priority
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1087:Mid‑priority
1065:WikiProjects
1048:
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1019:False claim?
986:
958:— Preceding
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798:Nick Boshaft
796:
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774:Rinconsoleao
771:
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587:Tom Lougheed
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1556:92.120.5.12
1520:Proprieties
1137:Mathematics
1128:mathematics
1084:Mathematics
931:—Preceding
866:—Preceding
817:—Preceding
811:null matrix
1572:Categories
1313:variables.
1543:Methods:
1340:algorithm
1053:is rated
972:contribs
960:unsigned
933:unsigned
868:unsigned
819:unsigned
1501:Sanitiy
1164:on the
1055:C-class
1025:section
1005:LeSnail
990:LeSnail
908:Derek M
604:Mct mht
574:Mct mht
1345:BLAS.
1061:scale.
964:Afbase
891:nbarth
846:nbarth
754:nbarth
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1242:over
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1560:talk
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