Geometric vs Algebraic Multiplicity

The gap between algebraic and geometric multiplicity is the single fact that decides whether a differential equation system $\dot{x} = Ax$ has solutions built purely from exponentials $e^{\lambda t} v$, or requires polynomial-times-exponential terms $t\, e^{\lambda t} v$ — this is exactly the content behind Diagonalizability Criterion and the construction of Generalized Eigenvectors for solving linear ODE systems with repeated eigenvalues. In numerical linear algebra, algorithms that assume diagonalizability (like naive eigenvector-based matrix power methods) become numerically unstable exactly near matrices with multiplicity gaps, motivating Schur decomposition and Jordan-form-aware numerical methods. In quantum mechanics, degenerate eigenvalues of an operator correspond to eigenspaces of dimension greater than one (geometric multiplicity $> 1$), and the question of whether a perturbation "splits" a degeneracy is precisely a question about how algebraic multiplicity distributes into new geometric multiplicities under perturbation. In graph theory and spectral graph theory, the geometric multiplicity of the eigenvalue $0$ for the graph Laplacian equals the number of connected components, a beautiful case where geometric multiplicity carries direct combinatorial meaning distinct from the raw polynomial factorization. In statistics, principal component analysis relies on symmetric (hence always diagonalizable) covariance matrices, where the two multiplicities always coincide — a fact from the Spectral Theorem that makes PCA well-posed without any Jordan-form complications.

Ask the tutor: Can you show me how to construct the full Jordan basis when geometric multiplicity is strictly less than algebraic multiplicity?

Picture the characteristic polynomial as a factory ledger: it records, with full precision, how many times each eigenvalue "should" appear, counted algebraically via factorization over $F$ or over an algebraic closure. The eigenspace, by contrast, is a physical measurement — you actually go into the vector space and count how many independent directions are truly fixed in direction (scaled, not rotated or sheared) by $T$. The ledger's count is a promise; the eigenspace's count is a delivery. The inequality $g(\lambda) \le a(\lambda)$ says the delivery can never exceed the promise, though it is allowed to fall short. The prototypical failure is a shear: the matrix $\begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}$ has characteristic polynomial $(x-\lambda)^2$, so $a(\lambda) = 2$, but its eigenspace is only the span of $\binom{1}{0}$, so $g(\lambda) = 1$. The polynomial "wanted" two independent eigendirections but the shear only delivers one; the missing direction gets absorbed into an off-diagonal entry that no eigenvector can see. This single $2\times 2$ block is the atomic building block of every gap between the two multiplicities, and it is exactly what generalized eigenvectors are built to repair.

Example 1 — a case of equality, with distinct block sizes and an asymmetric setup. Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be given in the standard basis by

This matrix is lower triangular, so its characteristic polynomial is

since the characteristic polynomial of a triangular matrix is the product of $(x - \text{diagonal entry})$ over all diagonal entries — a direct consequence of the fact that $xI - A$ is itself triangular, and the determinant of a triangular matrix is the product of its diagonal entries. So the eigenvalues are $\lambda_1 = 5$ with $a(5) = 1$, and $\lambda_2 = 3$ with $a(3) = 2$. For $\lambda_1 = 5$: solve $(A - 5I)v = 0$:

Row reduce: $R_2: v_1 = 2v_2$. Substitute into $R_3$: $2(2v_2) + 4v_2 - 2v_3 = 0 \Rightarrow 8v_2 = 2v_3 \Rightarrow v_3 = 4v_2$. Free parameter $v_2 = t$ gives $v = t(2,1,4)$. So $\dim E_5 = 1 = a(5)$: geometric multiplicity matches, trivially, since $a(5)=1$ forces $g(5) = 1$ by the theorem's inequality $1 \le g(5) \le a(5) = 1$. For $\lambda_2 = 3$: solve $(A - 3I)v = 0$:

Row 1 forces $v_1 = 0$; row 3 then forces $4v_2 = 0$, i.e. $v_2 = 0$; $v_3$ is free. So $E_3 = \operatorname{span}\{(0,0,1)\}$, giving $\dim E_3 = 1$. Here $g(3) = 1$ but $a(3) = 2$: the inequality is strict. This single example already shows both a tight case ($\lambda=5$) and a strict gap ($\lambda = 3$) inside one matrix, confirming that the theorem's inequality is the correct general statement — not always equality, not always strict. Since $g(3) + g(5) = 1 + 1 = 2 < 3 = \dim V$, this operator is not diagonalizable, by the corollary above (the eigenvectors found span only a $2$-dimensional subspace, insufficient to form a basis of $\mathbb{R}^3$).

Example 2 — a $4\times 4$ operator with a genuine multiplicity-3 eigenvalue and asymmetric geometric deficiency. Let $T: \mathbb{R}^4 \to \mathbb{R}^4$ be represented by

This matrix is block lower-triangular with an upper-triangular $3\times 3$ block (eigenvalue $2$, algebraic multiplicity $3$ from the diagonal) and a $1\times 1$ block (eigenvalue $5$). Formally,

computed via cofactor expansion along the last column, which has only two nonzero entries, reducing to the determinant of the leading $3\times 3$ upper-triangular block times $(x - 5)$. So $a(2) = 3$ and $a(5) = 1$. Compute $E_2 = \ker(A - 2I)$:

Row 1: $v_2 = 0$. Row 2: $v_3 = 0$. Row 4: $v_3 + 3v_4 = 0 \Rightarrow v_4 = 0$ (using $v_3 = 0$). So the only constraints force $v_2 = v_3 = v_4 = 0$, with $v_1$ free. Hence $E_2 = \operatorname{span}\{(1,0,0,0)\}$, so $g(2) = 1$. Here $a(2) = 3$ while $g(2) = 1$: a gap of size $2$. This shows the inequality can be strict by an arbitrarily large margin as the algebraic multiplicity grows — the deficiency $a(\lambda) - g(\lambda)$ is not bounded by any universal constant; it can be as large as $a(\lambda) - 1$. For $\lambda = 5$, the inequality forces $g(5) = 1$ automatically since $1 \le g(5) \le a(5) = 1$; direct computation of $\ker(A - 5I)$ confirms $E_5 = \operatorname{span}\{(0,0,0,1)\}$, one-dimensional, consistent. Total geometric multiplicity is $g(2) + g(5) = 1 + 1 = 2 < 4 = \dim V$, so $T$ is again not diagonalizable — the deficiency at $\lambda = 2$ alone kills diagonalizability, foreshadowing that this defect is exactly what Generalized Eigenvectors are constructed to recover, via a Jordan-type basis rather than a pure eigenbasis.

The theory of eigenvalues and their multiplicities crystallized through the joint efforts of several 19th-century figures working on quadratic forms, determinants, and the classification of linear substitutions, with the decisive structural result — how far a matrix can be from diagonal even when its characteristic polynomial has repeated roots — settled by Camille Jordan.