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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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[[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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