Talk:Persymmetric matrix

[Untitled]

Need to generate a picture to go with the words. n3vln 15:42, 8 April 2007 (UTC)[reply]

I don't remember ever seeing Muir's definition before (I'm used to another definition, which I added). Do you know whether Muir's definition is still being used? -- Jitse Niesen (talk) 04:47, 9 April 2007 (UTC)[reply]

Until I saw the term in an unresolved internal link, I had never seen it at all. I could not find the term in any of my other matrix algebra books. The persymmetric definition may these days be implicitly embedded in other names. The trend in Muirs treatise seems to be that mathematicians of that era defined a peculiar flavor of matrix and then investigated all the properties of that particular flavor ... the modern day approach appears more integrated. We'd have to revisit the context of the original entry in the history section of the determinant to understand the definition of the matrix under discussion. n3vln 12:13, 9 April 2007 (UTC)[reply]

Aha, you came here via determinant. I had a look and that section was added in this edit by User:Recentchanges who has an interesting list of contributions but hasn't edited since February 2004. It will be hard to track this user down!

It's likely that Muir used "persymmetric determinant" in the same sense as Sylvester, which is the context that determinant mentions the term. On the other hand, Toeplitz matrix refers to "persymmetric matrix" in the meaning that I added. So my best guess is that the term changed meaning somewhere in the twentieth century. Possibly it went out of fashion in the first half of the century, and resurfaced with another meaning in the 70s in the context of numerical analysis. Hmm, I'm starting to get interested! -- Jitse Niesen (talk) 13:07, 9 April 2007 (UTC)[reply]

Nosing around on MathSciNet, I found that "persymmetric" in Muir's meaning is used in Mays and Wojciechowski, "A determinant property of Catalan numbers", Discrete Math. 211 (2000), pp. 125-133. That's recent. The review starts with "In this paper a Hankel matrix is called a persymmetric matrix, but the idea is the same: constant entries on each skew diagonal." Lo and behold, we have an article Hankel matrix and that's indeed Muir's persymmetric matrix. So, what should we do now? -- Jitse Niesen (talk) 13:33, 9 April 2007 (UTC)[reply]

I spent a some time reviewing JSTOR articles. I found a 1931 paper by Aitken that uses the Muir definition. In google searches I many times ran across the phrases persymmetric Hankel or Hankel (persymmetric). Pretty clearly the modern literature defines persymmetric as defining symmetry around the minor diagonal (southwest to northeast), while Hankel appears to be the new designation for Muir persymmetric. So clearly there's the old and the new. I'm curious where the new definition started. Next step is to check Golub text for relevant bibliography entries. This might be an engineering field specific issue - I'll also check AMS sources. It's a tempest in a teapot, but it's interesting.

I would recommend promoting the second definition to the first, and making the use of persymmetric as synonymous with Hankel be secondary, esp. as there is now a separate article for that. Hardmath 17:27, 28 August 2007 (UTC)[reply]

Yes, I agree, so I went ahead. -- Jitse Niesen (talk) 02:13, 29 August 2007 (UTC)[reply]