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Please can you describe in layman's term, what does it actually mean by a "Characteristic Polynomial"?

Is it a property only of Matrices?

What does it describe about a Matrix, that is, what can we know about a Matrix from its "Characteristic Polynomial"?

muaddib
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  • The polynomial doesn't have much inherent significance. The eigenvalues are its roots, which have great significance; you can find plenty of resources discussing what the eigenvalues mean. – Ian Jun 25 '15 at 00:22
  • The characteristic polynomial of $M$ is $p(x) = \det(Ix - M)$. It's only defined for matrices, although there are numerous other unrelated things also called the "characteristic polynomial" in different contexts. – Jair Taylor Jun 25 '15 at 00:29
  • @Ian: The characteristic polynomial and its roots go hand in hand, so I don't see how one could be more significant than the other. In any case, the coefficients of the characteristic polynomial can be useful as well, especially for the Laplacian of a graph where they give the number of spanning forests (although this is too far afield to be an answer to the OP.) – Jair Taylor Jun 25 '15 at 00:31
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    @JairTaylor At least in terms of actually calculating the eigenvalues, the characteristic polynomial is usually pretty much useless except in the case $n=2$ where we have the quadratic formula. Maybe there are some applications of the coefficients elsewhere, but not really for finding eigenvalues. (Actually, polynomial root finding is usually made easier by reducing it to the eigenvalue problem for an appropriate matrix.) – Ian Jun 25 '15 at 00:32
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    See also http://math.stackexchange.com/questions/36651/interpreting-the-cayley-hamilton-theorem. – lhf Jun 25 '15 at 00:49
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    @Ian It is true that in terms of numerically finding eigenvalues of large matrices, the best methods do not go through the characteristic polynomial. But there's more to life than numerical methods, and in many cases exact symbolic computation using the characteristic polynomial can be useful. – Robert Israel Jun 25 '15 at 00:58
  • What "Layman's terms means" depends on what you know. Can we assume that you know what the determinant is? – Ben Grossmann Jun 25 '15 at 01:19
  • There are some properties which you get from the characteristic polynomial for example a matrix is nilpotent iff characteristic polynomial = $\lambda^n$ – Ainsley Pullen Jun 25 '15 at 07:45

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