Eigenvalue and eigenvector

For a linear transformation $T(\vec{x})=A\vec{x}:{\mathbb{R}}^n\to {\mathbb{R}}^n$; a non-zero vector $\vec{v}\in {\mathbb{R}}^n$ is an eigenvector of $A$ if:

$$A\vec{v}=\lambda \vec{v}$$

for some scalar $\lambda$, which is called the eigenvalue associated with eigenvector $\vec{v}$.

We can compute numerically:

• In NumPy, we use np.linalg.eig, which outputs a named tuple, with attributes eigenvalues and eigenvectors.
• In MATLAB, we use [V,D] = eig(A), which outputs a diagonal matrix of eigenvalues D and a matrix with corresponding eigenvectors V.

Definitions

We also define a basis $\{{\vec{v}}_1,\dots ,{\vec{v}}_n\}\in {\mathbb{R}}^n$ as an eigenbasis for $A$ if the vectors are eigenvectors of $A$. A handy tool is the characteristic polynomial, where $\lambda$ is an eigenvalue of $A$ if and only if $\text{det}(A-\lambda I_n)=0$.

Theorems

• Eigenvalues which correspond to distinct eigenvectors are linearly independent.
• If $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ has $n$ distinct eigenvalues, then $T$ is diagonalisable.
• The trace of $A$ is the sum of its eigenvalues, i.e.:

$$\text{tr}(A)=\sum _{i=1}^n{\lambda }_i$$

Computing the eigenvalues/eigenvectors of a matrix is doable with diagonalisation. Note that any real symmetric matrix can be decomposed into:

$$A=Q\Lambda Q^T$$

where $Q$ is composed of the eigenvectors of $A$, and $\lambda$ is a diagonal matrix.

Multiplicity

We define two helpful metrics:

Algebraic multiplicity: how many times an eigenvalue occurs, denoted $\text{almu}(\lambda )$. Geometric multiplicity: the dimension of the eigenspace $E_{\lambda }=\text{ker}(A-\lambda I_n)$, denoted $\text{gemu}(\lambda )=\text{dim}(E_{\lambda })=\text{nullity}(A-\lambda I_n)=n-\text{rank}(A-\lambda I_n)$.

We also observe by theorem, $\text{gemu}(\lambda )\le \text{almu}(\lambda )$.