Properties of a special block matrix

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Discussion Overview

The discussion revolves around the properties of a specific block matrix defined as R, which is composed of real matrices N and S, where N is symmetric and S is skew-symmetric. Participants explore theoretical aspects, potential applications, and relationships to other matrix forms, particularly focusing on eigenvalues and determinants.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant notes that the matrix R is symmetric, implying that its eigenvalues are real, but seeks further insights into its properties.
  • Another participant suggests that since any matrix can be expressed as a sum of symmetric and skew-symmetric parts, the specific forms of N and S may not contribute additional properties to R.
  • A participant mentions exploring a different matrix type and posits a relationship between the eigenvectors of that matrix and those of R, suggesting that the structure of eigenvectors might be influenced by the matrix A.
  • There is a query regarding the determinant of R, with one participant proposing that it may relate to the determinant of A, specifically questioning if det(R) equals det(-A^T A) or det(A^T A).
  • Another participant asserts that the determinant of R could be expressed as det(R) = det(A)^2, although this is presented as a possibility rather than an established fact.

Areas of Agreement / Disagreement

Participants express differing views on the contributions of the matrices N and S to the properties of R, and there is no consensus on the exact relationship of the determinant of R to A. The discussion remains unresolved regarding the implications of these properties.

Contextual Notes

Some assumptions about the matrices N and S, particularly regarding their specific forms and properties, remain unexamined. The implications of the eigenvector relationships and determinant calculations are also not fully resolved.

Who May Find This Useful

Readers interested in matrix theory, particularly in the properties of block matrices, eigenvalues, and determinants, may find this discussion relevant.

ekkilop
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Hi folks!

I've encountered the matrix below and I'm curious about its properties;

R=<br /> \begin{pmatrix}<br /> 0 &amp; N-S\\<br /> N+S &amp; 0<br /> \end{pmatrix}<br />

where R, N and S are real matrices, R is 2n by 2n, N is n by n symmetric and S is n by n skew-symmetric.

Clearly R is symmetric so the eigenvalues are real, but what else can be said about a matrix of this type? I checked through some literature but didn't really know what to look for. Surely the form is simple enough that it should have been studied.

In a special case, the elements of the rows of the matrix N+S sum to zero. Could this affect the properties somehow?

Any ideas would be much appreciated!
 
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Any matrix can be written as a symmetric plus a skew symmetric, so all you really have is a matrix of the form

R=<br /> \begin{pmatrix}<br /> 0 &amp; A\\<br /> A^t &amp; 0<br /> \end{pmatrix}<br />

The N and S aren't adding anything.

What are you doing with this matrix? Is there a specific type of problem you are trying to solve for example?
 
That's a fair point.
I was playing around with a different matrix - Hermitian and also symmetric about the anti-diagonal. Turns out that the eigenvectors are closely related to the eigenvectors of the matrix R above so I was curious about their structure. It seems reasonable that the upper and lower half of the eigenvectors should be closely related but the form largely depends on A I suppose.
But the form of R seems particularly neat so I thought perhaps it had some other interesting properties. Perhaps one could say what the determinant should be? Is it generally true that
det(R)=det(-A^{T}A)
 
ekkilop said:
Is it generally true that
det(R)=det(A^{T}A) [\tex]
<br /> That should be true for all block matrices (maybe with a different sign).<br /> As determinants are multiplicative, this can be simplified to det(R)=det(A)^2.
 
Thank you!
I think I shall have to return to the drawing board for a closer investigation :)
 

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