Eigenvalues & Eigenvectors

Definitions

Characteristic Polynomial

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

$$p(\lambda)=|A-\lambda I|$$

Eigenvalues

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

Note

• Product of the eigenvalues is equal to the determinant of the matrix
• Sum of the eigenvalues is equal to the trace of a matrix
• If $\lambda$ is an eigenvalue of $A$, then $\lambda^2$ is an eigenvalue of $A^2$
• $A$ and $A^T$ have the same eigenvalues.

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:

$$\frac{1}{k}=\sqrt{\sum_{i=1}^n X_i^2}$$

Norm

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

$$||W|| = \sqrt{\sum_{i=1}^n w_i^2}$$

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 $\lambda$.

Spectrum

Set of all eigenvalues.

Spectral Radius

$$R=\max\Big\{|\lambda_i|\; where \;\lambda_i \in \text{Spectrum}\Big\}$$

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:

$$k_1X_1+k_2X_2+k_3X_3+\dots+k_nX_n=0 \implies k_1=k_2=k_3=\dots=k_n=0$$

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.