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Peter Cudmore
Peter Cudmore
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I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that

$A = D + J$

Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.
When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.

So my question is this:
If $D$ has non-identical values (specifically, non-identical imaginary components), is there an elementary way to compute the eigenvalues of $A$ ?

The problem comes from linearising about the origin of a system of $n$ near identical coupled resonators. $D$ relates to the behaviour of each resonator, $A$$J$ relates to the coupling process.

I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that

$A = D + J$

Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.
When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.

So my question is this:
If $D$ has non-identical values (specifically, non-identical imaginary components), is there an elementary way to compute the eigenvalues of $A$ ?

The problem comes from linearising about the origin of a system of $n$ near identical coupled resonators. $D$ relates to the behaviour of each resonator, $A$ relates to the coupling process.

I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that

$A = D + J$

Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.
When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.

So my question is this:
If $D$ has non-identical values (specifically, non-identical imaginary components), is there an elementary way to compute the eigenvalues of $A$ ?

The problem comes from linearising about the origin of a system of $n$ near identical coupled resonators. $D$ relates to the behaviour of each resonator, $J$ relates to the coupling process.

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Peter Cudmore
Peter Cudmore
  • 293
  • 3
  • 10

Eigenvalues of the sum of a diagonal and a unit matrix

I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that

$A = D + J$

Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.
When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.

So my question is this:
If $D$ has non-identical values (specifically, non-identical imaginary components), is there an elementary way to compute the eigenvalues of $A$ ?

The problem comes from linearising about the origin of a system of $n$ near identical coupled resonators. $D$ relates to the behaviour of each resonator, $A$ relates to the coupling process.