Is it possible to construct a matrix with non-zero off-diagonal entries whose eigenvalues are nonetheless equal to its diagonal entries?

EDIT: @ajotataxe pointed out that this holds for triangular matrices. My follow up is - is the converse true? If the Eigen values are the diagonals, does it have to be triangular?


marked as duplicate by Jean Marie, erfink, Shailesh, Daniel W. Farlow, Leucippus May 24 '17 at 1:24

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  • $\begingroup$ Do you mean "non-zero entries off its diagonals"? (Otherwise the question is kind of trivial.) $\endgroup$ – Kenny Wong May 23 '17 at 17:28
  • $\begingroup$ Yes, non-zero entries off its diagonals. $\endgroup$ – Rohit Pandey May 23 '17 at 17:29
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    $\begingroup$ A triangular matrix? $\endgroup$ – ajotatxe May 23 '17 at 17:42
  • $\begingroup$ Right, triangular is true.. Thanks! Is it only possible for triangular? Let me edit the question. $\endgroup$ – Rohit Pandey May 23 '17 at 17:49
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    $\begingroup$ What would be truly surprising is a symmetric real matrix that is not diagonal and whose eigenvalues are on the diagonal. $\endgroup$ – Giuseppe Negro May 23 '17 at 17:55

Yes:$$\begin{pmatrix}0&\frac12&\frac12\\1&1&1\\1&-1&-1\end{pmatrix}.$$Its eigenvalues are $0$, $1$ and $-1$.

  • $\begingroup$ What Eigen vectors do you get for 1 and -1? $\endgroup$ – Rohit Pandey May 23 '17 at 17:53
  • $\begingroup$ Also, can you provide some intuition for how you constructed it? $\endgroup$ – Rohit Pandey May 23 '17 at 17:55
  • $\begingroup$ Ok, I got the Eigen vectors - [0.3015,0.3015,0] and [0.9,0.3015,-0.707]. How did you construct it though? $\endgroup$ – Rohit Pandey May 23 '17 at 17:59
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    $\begingroup$ I started with the matrix $\left(\begin{smallmatrix}0&a&b\\c&1&d\\e&f&-1\end{smallmatrix}\right)$. Its characteristic polynomial is $-x^3+a c x+b e x+d f x+x+a c-b e+a d e+b c f$. So, I searched for numbers $a$, $b$, $c$, $d$, $e$, and $f$ such that $ac+be+de=0$ and that $a c - b e + a d e + b c f=0$. In order to do that, I solved the system in order to $a$ and $b$ and then I chose more or less random values for $c$, $d$, $e$, and $f$. $\endgroup$ – José Carlos Santos May 23 '17 at 18:22

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