Characteristic polynomial of a hollow matrix

Asked Oct 13 '22 at 16:19

Active Oct 13 '22 at 16:48

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I'm searching a way to calculate the eigenvalues of the matrix

$$\begin{pmatrix} 0 & 1 & \dots & 1 & 1 \\ 1 & 0 & \dots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \dots & 0 & 1 \\ 1 & 1 & \dots & 1 & 0 \end{pmatrix}$$

Is it smarter to analyze the difference ${\bf J} - {\bf I}_{n \times n}$, where $\bf J$ is the matrix full with ones and ${\bf I}_{n \times n}$ the identity matrix?

edited Oct 13 '22 at 16:48

Rodrigo de Azevedo

asked Oct 13 '22 at 16:19

vitalmath

1

Yes indeed! The eigenvalues of $A+cI$ are always $c$ plus the eigenvalues of $A$ (with the same eigenvectors); and here the eigenvalues of $J$ are easy to determine. – Greg Martin Oct 13 '22 at 16:23
Related – Rodrigo de Azevedo Oct 13 '22 at 16:33
Related – Rodrigo de Azevedo Oct 13 '22 at 16:35
Related – Rodrigo de Azevedo Oct 13 '22 at 16:36

Characteristic polynomial of a hollow matrix

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