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== Inverses/products of triangular matrices ==
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In the article's [https://en.wikipedia.org/w/index.php?title=Triangular_matrix&oldid=761938496#Simultaneous_triangularisability section] about simultaneous triangularisability is claimed that
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{{quotation|The fact that commuting matrices have a common eigenvector can be interpreted as a result of [[Hilbert's Nullstellensatz]]: commuting matrices form a commutative algebra <math>K[A_1,\ldots,A_k]</math> over <math>K[x_1,\ldots,x_k]</math> which can be interpreted as a variety in ''k''-dimensional affine space, and '''the existence of a (common) eigenvalue''' (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 [[algebra representation]] of the polynomial algebra in ''k'' variables.}}
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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 BLAS. [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 17:53, 4 February 2018 (UTC)
== Chapter: Forward and back substitution ==
The algorithm for forward substitution as it is now, assumes that <math>l_{i,i}\neq 0</math>, which isn't given in general.
Furthermore, the algorithm isn't well defined for triangular matrices that don't have a staircase form, e.g.
<math>\begin{pmatrix}0&1&1\\0&1&2\\0&0&0\end{pmatrix}</math>.
I'd therefore argue that this chapter, though generally formulated for triangular matrices, would be a better fit for [[Row_echelon_form]].
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)
[[User:Sanitiy|Sanitiy]] ([[User talk:Sanitiy|talk]]) 17:42, 21 December 2019 (UTC)
== Proprieties, Special Forms, Methods ==
On '''Proprieties''' 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.
http://homepages.warwick.ac.uk/~ecsgaj/matrixAlgSlidesC.pdf
Each entry on the main diagonal of L-1 is equal to the reciprocal of the corresponding entry on the main diagonal of L.
https://www.statlect.com/matrix-algebra/triangular-matrix
A Venn diagram of types of triangular matrices would be helpful in '''Special forms'''
(((Identity) Diagonal)Atomic)Uni-triangular)Non-Singular, Singular(Strictly triangular)
and bi-diagonal
Some algorithms can be added from:
"Stability of Methods for Matrix Inversion"
http://www.netlib.org/lapack/lawnspdf/lawn27.pdf
Methods:
''Unblocked method''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 ''blocked methods'' author presents 1B, 2B, 2C methods.
https://epubs.siam.org/doi/abs/10.1137/0119075?journalCode=smjmap [[Special:Contributions/92.120.5.12|92.120.5.12]] ([[User talk:92.120.5.12|talk]]) 11:56, 7 August 2023 (UTC)
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