Rank of a matrix with identical diagonal, upper triangle, and lower triangle

Asked Dec 27 '23 at 08:07

Active Dec 27 '23 at 17:16

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I'm trying to figure out when the matrix below is full rank or non-singular. All the elements in the upper triangle (non-diagonal) are equal to $a$, elements in the lower triangle are equal to $c$, and diagonal elements are all $b$. Also, $a > b > c$, and $b$ can be assumed not to be zero if it helps.

$$ \begin{bmatrix} b & a & a & \dots & a \\ c & b & a & \dots & a \\ c & c & b & \dots & a \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c & c & c & \dots & b \end{bmatrix} $$

Is such matrix always full rank? If not, can we make it full rank by making extra assumptions on the elements?

I appreciate any help with this!

edited Dec 27 '23 at 17:16

Rodrigo de Azevedo

asked Dec 27 '23 at 08:07

Nariman

1

To answer a part of your question: can it be made full rank? An obvious choice is $c=0$. – Anurag A Dec 27 '23 at 08:12
@AnuragA Right right; Thanks. I'm still hoping for full rank without extra assumption though. – Nariman Dec 27 '23 at 08:18
Consider the $2{\times}2$ case with $b^2=ac$. – quasi Dec 27 '23 at 09:39
If $a = c$, then take a look at this – Rodrigo de Azevedo Dec 27 '23 at 17:22
@RodrigodeAzevedo Thanks, that can give me some ideas. In my case though, $a$ is greater than $c$. – Nariman Dec 28 '23 at 05:32

Rank of a matrix with identical diagonal, upper triangle, and lower triangle

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