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I am interested in real symmetric matrices of the form:

$$\mathbf{M} = \begin{bmatrix} a & t & t & \cdots & t \\ t & a & t & \cdots & t \\ t & t & a & \cdots & t \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ t & t & t & \cdots & a \text{ } \text{ } \\ \end{bmatrix}.$$

This is a simple matrix form where the diagonal elements have a constant value $a \in \mathbb{R}$ and the off-diagonal elements have a constant value $t \in \mathbb{R}$. Some useful special cases of this matrix form are the centering matrix and the equicorrelation matrix. (This matrix form is also a particular case of the Toeplitz matrix, but it is much simpler than that general form.)

Question: Does this matrix form have a name?

Rodrigo de Azevedo
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Ben
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  • see http://math.stackexchange.com/questions/1414690/matrix-with-all-1s-diagonalizable-or-not/1414925#1414925 or many others. Your matrix is $t$ times the matrix with all $1,$ then add $(a-t)I.$ The eigenvalues are easy, $n-1$ copies of one thing and a singleton of another. – Will Jagy Feb 22 '19 at 00:22
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    @Will Jagy: It is unclear to me how that answer has any bearing at all on my question. I am aware of the properties of this matrix. What is the name of this matrix? – Ben Feb 22 '19 at 00:36
  • It is also a special case of a symmetric circulant matrix. Not all matrix families have names yet. – Somos Feb 22 '19 at 01:10
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    @Somos: Hence the purpose of asking the question. – Ben Feb 22 '19 at 01:31

2 Answers2

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In old statistics literature, these are sometimes called completely symmetric matrices. I have made a light effort to revive the term in my writing or teaching (which is usually a combinatorial context).

Peter Dukes
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This class of matrices is now examined in detail in O'Neill (2020) where they are called "double-constant" matrices. Although these matrices have been mentioned in some books on linear algebra in statistics, they do not appear to have had any existing name in the literature prior to this paper. As noted in the paper, this class of matrices includes the centering matrix and the equicorrelation matrix, both used widely in statistical applications.

Ben
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