Talk:Triangular matrix

Inverses/products of triangular matrices

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)

I've added a paragraph about triangular matricies preserving form. Tom Lougheed 02:32, 12 August 2007 (UTC)

Removed false claim

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 ${\begin{bmatrix}1&&\\2&1&\\3&4&1\\\end{bmatrix}}{\begin{bmatrix}1&&\\-2&1&\\-3&-4&1\\\end{bmatrix}}={\begin{bmatrix}1&&\\0&1&\\-8&0&1\\\end{bmatrix}}\neq \mathbf {I}$

Tom Lougheed 01:17, 12 August 2007 (UTC)

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)

The claim is still false. Look at the counter example. Tom Lougheed 01:27, 12 August 2007 (UTC)

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)

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)

Other false claim

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 first matrix and with the second matrix that permutes these blocks.

Algebra of upper triangular matrices

Is there a standard notation for the algebra/ring of upper triangular matrices?--129.70.14.128 (talk) 23:09, 16 December 2007 (UTC)

You can use ${\mathfrak {b}}$ for “Borel subalgebra”, and for strictly upper triangular, ${\mathfrak {n}}$ for “Nilpotent”. This is a bit heavy duty (Lie algebra notation), but is a standard.

—Nils von Barth (nbarth) (talk) 08:22, 2 December 2009 (UTC)

Quasi-triangular matrices

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)

Applied Numerical Linear Algebra, James W. Demmel, 1997, copyright SIAM, page 147.

"THEOREM4.3. Real Schur canonical form. 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 eigenvalues, and the 2-by-2 blocks to complex conjugate pairs.

Nick Boshaft (talk) 00:48, 28 April 2016 (UTC)

null matrix

i wanna know if a null matrix would be called an upper triangular or lower triangular. —Preceding unsigned comment added by 117.200.51.168 (talk) 07:29, 17 March 2009 (UTC)

You’d be better served to ask questions at the Wikipedia:Reference desk, as that is far more watched than individual article pages. In the event, an all zero square matrix is both upper triangular and lower triangular.

—Nils von Barth (nbarth) (talk) 08:26, 2 December 2009 (UTC)

square matrix

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. —Preceding unsigned comment added by 158.64.77.124 (talk) 16:35, 9 November 2009 (UTC)

“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.

—Nils von Barth (nbarth) (talk) 08:24, 2 December 2009 (UTC)

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 context of LU Decompositions or is this something that is otherwise rare other than this book?

Derek M (talk) 08:53, 17 March 2018 (UTC)

Forward and Back Substitution

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)

triangularisability

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

This article is highly disorganized

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)

I've worked a bit on the first half now. The second half is untouched. LeSnail (talk) 03:34, 19 March 2012 (UTC)

False claim?

In the article's section about simultaneous triangularisability is claimed that

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 $K[A_{1},\ldots ,A_{k}]$ over $K[x_{1},\ldots ,x_{k}]$ 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.

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)

algorithm

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)

Chapter: Forward and back substitution

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)

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 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)