Multiplicity of eigenvalue

Algrvraic multiplicity

Definition

The algrbraic multiplicity of an eigenvalue $\lambda$ of $A$ is the multiplicity of $\lambda$ as roots of the characteristic polynomial $p_A(x)$.

• If $A$ is $n\times n$, then it has $n$ eigenvalues counting algebraic multiplicities.

Prop

$\text{trace}(A)=\sum {\lambda }_i,\text{det}(A)=\prod {\lambda }_i$

• $A$ is invertible iff it has no zero eigenvalue

Prop

If $A=\left[\begin{array}{cc}B& \ast \\ O& C\end{array}\right]$, then the characteristic polynomials have the relation $p_A(x)=p_B(x)p_C(x)$.

Schur Decomposition

For any square matrix $A$, we can find an invertible matrix $X$ s.t. $A=XT{X}^{-1}$ where $T$ is upper triangular and the diagonal entries of $T$ are exactly the eigenvalues of $A$.

• If $A$ has eigenvalues ${\lambda }_1,\dots ,{\lambda }_n$, then for any polynomila $p(x)$, the matrix $p(A)$ has eigenvalues $p({\lambda }_1),\dots ,p({\lambda }_n)$.

Geometric multiplicity

Definition

The eigenspace of an eigenvalue $\lambda$ for a matrix $A$ is $\text{Ker}(A-\lambda I)$. We say the geometric multiplicity of $\lambda$ is $\text{dim Ker}(A-\lambda I)$.

• The eigenspaces of different eigenvalues are linearly independent.

Prop

An $n\times n$ matrix $A$ is diagonalizable iff its gemometric multilplicities add up to $n$.

Let $\lambda$ be an eigenvalue of $A$. Then $m_g(\lambda )\le m_a(\lambda )$.

A matrix $A$ is diagonalizable iff for each eigenvalue $\lambda$, its algebraic multiplicity equals to its geometric multiplicity.