Eigenvalues & Eigenvectors

Definitions

Characteristic Polynomial

Let $A$ be a $n\times n$ matrix.

Eigenvalues

Roots of the equation $p(\lambda) = 0$ are the eigenvalues of $A$.

Eigenvectors

The column vectors satisfying the equation $(A-\lambda_i I)X_i$.

Normalized Eigenvectors

An eigenvector with the magnitude (norm) of $1$. Normalizing factor $k$ of any eigenvector is:

Norm

Norm of a column or row matrix $W_{n\times n}$ is denoted by $||W||$ and defined as:

Algebraic Multiplicity

If the characteristic polynomial consists of a factor of the form $(\lambda - \lambda_i)^r$ and $(\lambda - \lambda_i)^{r+1}$ is not a factor of the characteristic polynomial then $r$ is the algebraic multiplicity of the eigenvalue $\lambda$. Denoted by $a_\lambda$.

Spectrum

Set of all eigenvalues.

Spectral Radius

Linear Independence of Eigenvectors

Suppose $X_1,X_2,X_3,\dots,X_n$ is a set of eigenvectors. $k_1,k_2,k_3,\dots,k_n$ is a set of scalars.

All those eigenvectors are independent iff:

For Special Matrices

Real Symmetric Matrix

Suppose $A$ is a symmetric matrix with all real entries. Then:

• The eigenvalues of $A$ are all real: $\forall \lambda \in S_A, (\lambda_i \in \mathbb{R})$
• The eigenvectors of $A$ (corresponding to distinct values of $\lambda$) are mutually orthogonal
• $A$ and $A^T$ have the same eigenvalues

Upper Triangular Matrix

The eigenvalues are the diagonal entries.