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Can you help me understand this statement:

An eigenvalue c has algebraic multiplicity $k$ if $(t-c)^k$ is the highest power of $(t-c)$ that divides the characteristic polynomial.

I am not sure, what does $t$ stand for. I have lifted this statement from the first statement under Algebraic Multiplicity heading from this link

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    $t$ is the variable in the characteristic polynomial. The characteristic polynomial is a *polynomial*, computed as $\det(tI-A)$, where $A$ is the matrix of the linear transformation. You may be used to different notation; just substitute the name of the variable you are used to.2012-06-17
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    Thanks I suspected as much. Is there a possibility you can sneak in some help in how to prove it.2012-06-17
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    How to prove what? It's a *definition*. There is nothing to prove.2012-06-17
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    It looks like the definition should read "... c has algebraic multiplicity k if k is the highest power of ...". Perhaps it's a typo...2012-06-17
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    @Andrew: How so? It says "$c$ has algebraic multiplicity $k$ if $(t-c)^k$ is the highest power of $(t-c)$ that divides the characteristic polynomial." It seems perfectly correct to me.2012-06-17
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    @ArturoMagidin You're right, I misread it. Thanks.2012-06-17

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The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial.

Then a root $c$ of $P(t)$ has multiplicity $\mu$ if $\mu$ is the highest integer such as $(t-c)^\mu$ divides $P(t)$.