Talk:Triangular matrix

message: 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 triangular.  We should probably state these properties more directly, and perhaps clean up the article in general.  --Rriegs 04:11, 5 May 2007 (UTC)
replies: text: I've added a paragraph about triangular matricies preserving form. Tom Lougheed 02:32, 12 August 2007 (UTC)
thread_title:  Inverses/products of triangular matrices 

message: I deleted a line from the article falsely claimed that
:"Indeed, we have
:: \mathbf{L}^{-1} =
\begin{bmatrix}
     1 &        &           &         &         & 0 \\
       & \ddots &           &         &         &   \\
       &        &         1 &         &         &   \\
       &        &-l_{i+1,i} &  \ddots &         &   \\
       &        &    \vdots &         &  \ddots &   \\
     0 &        &  -l_{n,i} &         &         & 1 \\
\end{bmatrix},
:i.e. the off-diagonal entries are replaced by their opposites."
Except for the first sub-diagonal, the inverse of an atomic lower triangluar is not quite as simple as reversing signs.
Consider this counter example:
: \mathbf{L} =
\begin{bmatrix}
     1 &   &    \\
     2 & 1 &    \\
     3 & 4 & 1  \\
\end{bmatrix}
Notice that
\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} \ne \mathbf{I}
Tom Lougheed 01:17, 12 August 2007 (UTC)
replies: text: 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. Mct mht 01:25, 12 August 2007 (UTC)
replies: text: The claim is still false. Look at the counter example. Tom Lougheed 01:27, 12 August 2007 (UTC)
replies: text: 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. Mct mht 01:34, 12 August 2007 (UTC)
replies: text: 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 \mathbf{L}_i 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. Tom Lougheed 02:32, 12 August 2007 (UTC)
thread_title:  Removed false claim 

message: 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? --Rinconsoleao (talk) 22:12, 28 February 2008 (UTC)
thread_title:  Quasi-triangular matrices 

message: 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 Ly = b.  No algorithm or equations are given for back sub of a given upper diagonal matrix.  —Preceding unsigned comment added by 67.239.155.230 (talk) 16:22, 4 March 2011 (UTC)
thread_title:  Forward and Back Substitution 

message: is triangularisability a word?  — Preceding unsigned comment added by Afbase (talk • contribs) 03:27, 11 September 2011 (UTC)
thread_title: triangularisability

message: 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! LeSnail (talk) 01:55, 19 March 2012 (UTC)
replies: text: I've worked a bit on the first half now. The second half is untouched. LeSnail (talk) 03:34, 19 March 2012 (UTC)
thread_title:  This article is highly disorganized 

message: In the article's section  about simultaneous triangularisability is claimed that 
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 I and 2I share common eigenvectors, but their eigenvalues are different. Saung Tadashi (talk) 23:06, 26 February 2017 (UTC)
thread_title:  False claim? 

message: 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. 67.198.37.16 (talk) 17:53, 4 February 2018 (UTC)
thread_title:  algorithm 

message: The algorithm for forward substitution as it is now, assumes that l_{i,i}\neq 0, 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.
\begin{pmatrix}0&1&1\\0&1&2\\0&0&0\end{pmatrix}.
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)
Sanitiy (talk) 17:42, 21 December 2019 (UTC)
thread_title:  Chapter: Forward and back substitution 

message: 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 methods “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 92.120.5.12 (talk) 11:56, 7 August 2023 (UTC)
thread_title:  Proprieties, Special Forms, Methods