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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. –  Arturo Magidin Jun 17 '12 at 20:07
    
Thanks I suspected as much. Is there a possibility you can sneak in some help in how to prove it. –  Soham Jun 17 '12 at 20:09
    
How to prove what? It's a definition. There is nothing to prove. –  Arturo Magidin Jun 17 '12 at 20:09
    
It looks like the definition should read "... c has algebraic multiplicity k if k is the highest power of ...". Perhaps it's a typo... –  Andrew Jun 17 '12 at 20:12
    
@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. –  Arturo Magidin Jun 17 '12 at 20:14

1 Answer 1

up vote 2 down vote accepted

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 $P(t)$.

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