📂Matrix Algebra

Algebraic and Geometric Multiplicities of Eigenvalues

Algebraic Multiplicity

For a matrix $A \in \mathbb{R}^{m \times m}$, the eigenvalue is defined as $\lambda$ that satisfies $\det (A - \lambda I ) =0$. The characteristic equation is a polynomial equation of degree $m$ with respect to $\lambda$, which can be expressed as

$$\det (A - \lambda I ) = (-1)^m \lambda ^m + c_{m-1} \lambda ^{m-1} + \cdots + c_{1} \lambda + c_{0} = 0$$

According to the Fundamental Theorem of Algebra, the characteristic equation has exactly $m$ roots, including complex numbers. Here, a root includes repeated roots, meaning that eigenvalues can be found with multiplicity. To focus on repeated roots, let’s express the characteristic equation in its factored form.

$$\det ( A - \lambda I) = c (\lambda - \lambda_1)^{a_1} (\lambda - \lambda_2)^{a_2} \cdots (\lambda - \lambda_k)^{a_k}$$

$$k \le m \\ \sum_{i=1}^{k} a_{i} = m$$

When expressed as above, the matrix $A$ has $k$ distinct eigenvalues, and $\lambda_{i}$ is repeated $a_{i}$ times. We define that the eigenvalue $\lambda_{i}$ has an algebraic multiplicity of $a_{i}$.

Geometric Multiplicity

On the other hand, let’s consider another explanation of eigenvalue in terms of its geometric meaning. Suppose that $\mathbf{x}_1, \mathbf{x}_2 \in \mathbb{C}^{m}$ is a solution to the matrix equation $A \mathbf{x} = \lambda_{i} \mathbf{x}$ for the eigenvalue $\lambda_{i}$ of matrix $A$. Then, the two vectors $\mathbf{x}_1, \mathbf{x}_2$ will be eigenvectors corresponding to the same eigenvalue $\lambda_{i}$. Of course, there can be infinitely many eigenvectors for one eigenvalue. This is geometrically because there can exist scaled versions of the eigenvector $\mathbf{x}$, represented as $\alpha \mathbf{x}$.

However, what if $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ are orthogonal to each other? They share the same eigenvalue but cannot represent each other by scaling due to their linear independence.

Let’s generalize this discussion.

$$S_{\lambda_{i}} = \left\{ x \in \mathbb{C}^{m} \ | \ A \mathbf{x} = \lambda_{i} \mathbf{x} \right\}$$

represents the set of all eigenvectors corresponding to the eigenvalue $\lambda_{i}$ of the matrix $A$. If we denote this set as $g_{i} = \dim S_{\lambda_{i}}$, then $g_{i}$ represents the number of types of eigenvectors that share the eigenvalue $\lambda_{i}$ but are orthogonal to each other. We define that the eigenvalue $\lambda_{i}$ has a geometric multiplicity of $g_{i}$.

See Also

Comparison between Algebraic and Geometric Multiplicity

Naturally, there is no guarantee that the algebraic and geometric multiplicities are typically the same. And if somewhere the term ‘multiplicity of an eigenvalue’ is used without further explanation, it most likely refers to the algebraic multiplicity.

One of the reasons for specifically defining geometric multiplicity (of course it can be adequately explained by the essence of mathematics) is due to its emergence in physics.

Degeneracy in Quantum Mechanics

Refers to the state where two different wave functions share the same eigenvalue. In physics, where ‘representing a determinant as a polynomial’ is not heavily emphasized, this situation implies geometric multiplicity. Just as in mathematics, where one cannot distinguish between eigenvectors with just the eigenvalue, in physics, it is impossible to differentiate wave functions just by the energy level.