📂Matrix Algebra

Dominant Eigenvalue

Definition¹

square matrix $A$ has distinct eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}$. Then $\lambda_{j}$ that satisfies the following is called the $A$’s dominant eigenvalue, and the corresponding eigenvector is called the $A$’s dominant eigenvector.

$$ | \lambda_{j} | \gt | \lambda_{i} | \quad \text{for all } i \ne j $$

Explanation

Since the $\lambda_{i}$ in the definition are distinct eigenvalues, the dominant eigenvalue, if it exists, is unique. A dominant eigenvalue may or may not exist depending on the matrix. For example, the eigenvalues of the matrix $A = \begin{bmatrix} -4 & 0 & 0 \\ 0 & -5 & 3 \\ 0 & -6 & 4 \end{bmatrix}$ are $-4$, $-2$, and $1$, so the dominant eigenvalue of $A$ is $-4$. On the other hand, the eigenvalues of the matrix $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 5 \\ 0 & 1 & -2 \end{bmatrix}$ are $-3$, $1$, and $3$, so $B$ has no dominant eigenvalue.