Talk:Triangular matrix

Inverses/products of triangular matrices

Latest comment: 18 years ago2 comments2 people in discussion

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)Reply

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

Removed false claim

Latest comment: 18 years ago5 comments2 people in discussion

I deleted a line from the article falsely claimed that

"Indeed, we have

${\displaystyle \mathbf {L} ^{-1}={\begin{bmatrix}1&&&&&0\\&\ddots &&&&\\&&1&&&\\&&-l_{i+1,i}&\ddots &&\\&&\vdots &&\ddots &\\0&&-l_{n,i}&&&1\\\end{bmatrix}},}$ 

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:

${\displaystyle \mathbf {L} ={\begin{bmatrix}1&&\\2&1&\\3&4&1\\\end{bmatrix}}}$ 

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

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

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)Reply

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

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)Reply

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 ${\displaystyle \mathbf {L} _{i}}$  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)Reply

Algebra of upper triangular matrices

Latest comment: 17 years ago1 comment1 person in discussion

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

Quasi-triangular matrices

Latest comment: 17 years ago1 comment1 person in discussion

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)Reply

null matrix

Latest comment: 16 years ago1 comment1 person in discussion

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)Reply

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.