I was wondering if there is a name for matrix like following:
It is like a symmetric matrix but to the other side. Is there a definition for this type of matrices?
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2$\begingroup$ See math.stackexchange.com/questions/378696/… $\endgroup$– GBaardinkCommented May 27, 2019 at 17:17
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3$\begingroup$ Possible duplicate of Name of a special matrix $\endgroup$– DanLewis3264Commented May 27, 2019 at 17:18
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3$\begingroup$ It's a Toeplitz matrix. $\endgroup$– user1551Commented May 27, 2019 at 17:18
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2$\begingroup$ @bounceback This question is not a duplicate of the linked "Name of special matrix" question, because "Name of special matrix" is not about Toeplitz matrices. The matrix given in this question has a structure which is not present in the other question. $\endgroup$– littleOCommented May 27, 2019 at 17:29
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1$\begingroup$ OK, perhaps I was thrown by the line 'It is like a symmetric matrix but to the other side', but I see your point. $\endgroup$– DanLewis3264Commented May 27, 2019 at 20:20
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2 Answers
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As @user1551 mentioned, this is a Toeplitz matrix:
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1$\begingroup$ I had taken the second definition of the OP "symmetric with respect to the second diagonal" and didn't pay attention to the example that was in fact more restrictive : all descending diagonals have the same entry... $\endgroup$ Commented May 27, 2019 at 19:14
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Disclaimer : I stick to your definition "It is like a symmetric matrix but to the other side". I don't take into account the supplementary fact that all descending diagonals contain the same entry that exists on your example but may not exist as well.
Up to my knowledge, this kind of matrix has no special name. Let us call them provisionally $T$-matrices.
It must be understood that all operations that can be done on $T$-matrices are literally "mirrored" into operations done on symmetric matrices $S$ through operation ;
$$T \to JTJ=JTJ^{-1}=S\tag{1}$$
with matrix $J$ defined by :
$$J:=\begin{pmatrix}0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0\end{pmatrix} \ \ \ \text{with} \ \ J^2=I \ \iff \ J^{-1}=J$$
(or the equivalent in $n$ dimensions).
Remark : (1) is a bijection, with inverse operation $T=JSJ$. Moreover, this relationship shows that $S$ and $T$ are conjugated ; thus in particular, they have the same eigenvalues.
answered May 27, 2019 at 17:21
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$\begingroup$ That can't be right. For example, the space of symmetric $4\times 4$ matrices has dimension $10$, while the space of $4\times 4$ T matrices has dimension $7$. $\endgroup$ Commented May 27, 2019 at 19:50
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$\begingroup$ @David C. Ullrich See the "disclaimer" I just added in front of my answer. $\endgroup$ Commented May 27, 2019 at 20:03
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1$\begingroup$ The term is persymmetric. I would say you answered the question as asked. The example is also Toeplitz, but that is too restrictive. $\endgroup$ Commented May 27, 2019 at 20:24
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$\begingroup$ @Rodrigo de Azevedo You are right : I had seen this term, but forgot it. $\endgroup$ Commented May 27, 2019 at 20:26
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$\begingroup$ Right. You say you used the definition "It is like a symmetric matrix but to the other side". I couldn't/cam't figure out exactly what that means... $\endgroup$ Commented May 28, 2019 at 12:56