Mathematik-Online lexicon: Example: Algebraic and Geometric Multiplicity

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	Mathematik-Online lexicon:
	Example: Algebraic and Geometric Multiplicity

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overview

The matrices

$\displaystyle A_1=\left(\begin{array}{rrr}4& 0 &0 \\ 0 & 4 & 0\\ 0 & 0 & 4\end{... ...3=\left(\begin{array}{rrr}4& 4 &0 \\ 3 & 4 & 6\\ 0 & -2 & 4\end{array}\right)$

have all the same characteristic polynomial

$\displaystyle (4-\lambda)^3=(4-\lambda)(5-\lambda)(3-\lambda)+4-\lambda=(4-\lambda)^3+12(4-\lambda)-12(4-\lambda)$

and, hence, the respective algebraic multiplicity is

. By means of the respective rank of the matrices

$\displaystyle (A_1-4E)=\left(\begin{array}{rrr}0& 0 &0 \\ 0 & 0 & 0\\ 0 & 0 & 0... ...=\left(\begin{array}{rrr}0& 4 &0 \\ 3 & 0 & 6\\ 0 & -2 & 0\end{array}\right)$

we see that

for

and

for

see also:

Algebraic and Geometric Multiplicity

automatisch erstellt am 25. 6. 2018