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Mathematics-Online course: Linear Algebra - Normal Forms - Eigenvalues and Eigenvectors

Algebraic and Geometric Multiplicity

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Let $ \lambda$ be an eigenvalue of matrix $ A$. The algebraic multiplicity $ m_{\lambda}$ of eigenvalue $ \lambda$ is defined as the multiplicity of $ \lambda$ as zero of $ \det(A - \lambda E_n)$.
The geometric multiplicity of eigenvalue $ \lambda$ is defined as the dimension $ d_{\lambda}$ of the eigenpace of $ \lambda$.
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 $ m_4=3$. 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 $ d_4=3$ for $ A_1$, $ d_4=2$ for $ A_2$ and $ d_4=1$ for $ A_3$.

  automatically generated 4/21/2005