|WikiProject Mathematics||(Rated C-class, Low-importance)|
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
- 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:
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 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
- You can use for “Borel subalgebra”, and for strictly upper triangular, 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)
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.
- 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)
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 184.108.40.206 (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)
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 220.127.116.11 (talk) 16:22, 4 March 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)
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 over 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)