Common Errors in Jordan Chain Calculations

Start with the Chain Equations

Jordan-chain calculations are especially vulnerable to errors that look plausible in a matrix: reversing a chain can reverse the off-diagonal pattern, a generalized eigenvector can be mistaken for an eigenvector, and a correct geometric multiplicity can be used to draw an unjustified conclusion about block sizes. The safest check is to return to the defining equations and test each vector directly.

Let \(T:V\to V\) be a linear map on a finite-dimensional vector space over \(\mathbb{C}\), and let \(\lambda\) be an eigenvalue. (The same discussion applies over \(\mathbb{R}\) when the eigenvalue and vectors lie in \(\mathbb{R}\).) A Jordan chain of length \(k\) for \(\lambda\), in the convention used here, is an ordered list \((\mathbf{v}_1,\dots,\mathbf{v}_k)\) satisfying

with \(\mathbf{v}_1\ne\mathbf{0}\). The first vector is an eigenvector. Each later vector is a generalized eigenvector: repeated application of \(T-\lambda\operatorname{id}_V\) eventually sends it to \(\mathbf{0}\). In particular, the equations imply \((T-\lambda\operatorname{id}_V)^j\mathbf{v}_j=\mathbf{0}\), while \((T-\lambda\operatorname{id}_V)^{j-1}\mathbf{v}_j=\mathbf{v}_1\ne\mathbf{0}\).

These equations are a useful diagnostic in their own right. If a proposed chain does not satisfy the recurrence for every index, it is not a chain in this order. If the recurrence holds, only the first vector is required to be an eigenvector; later vectors usually are not.

Why the Standard Jordan Block Has a Superdiagonal

The Jordan block convention places ones immediately above the diagonal. This pattern follows from the chain equations and the order of the basis; it is not a separate rule to memorize.

Proof. Write the matrix of \(T\) in the stated basis by expressing the image of each basis vector as a column of coordinates. The chain equations give \(T\mathbf{v}_1=\lambda\mathbf{v}_1\). For each \(j\) with \(2\le j\le k\), they give \(T\mathbf{v}_j=\lambda\mathbf{v}_j+\mathbf{v}_{j-1}\). Therefore column \(1\) has \(\lambda\) in its first entry and zeros elsewhere. Column \(j\) has \(\lambda\) in row \(j\), \(1\) in row \(j-1\), and zeros elsewhere. Those entries are exactly \(\lambda\) on the diagonal and \(1\) immediately above it. \(\square\)

Reversing the basis order changes the displayed pattern. In the same example, use \((\mathbf{e}_3,\mathbf{e}_2,\mathbf{e}_1)\). Then \(\mathbf{A}\mathbf{e}_3=2\mathbf{e}_3+\mathbf{e}_2\), \(\mathbf{A}\mathbf{e}_2=2\mathbf{e}_2+\mathbf{e}_1\), and \(\mathbf{A}\mathbf{e}_1=2\mathbf{e}_1\). Their coordinates in the reversed basis form

This is a lower-triangular pattern, not the standard superdiagonal Jordan block under the stated convention. The underlying map has not changed; only the basis order has. The reversed list is not ordered as a chain under the defining recurrence above. A common error is to reverse the vectors but still label the resulting matrix the standard Jordan block without checking which convention is being used.

Generalized Eigenvectors Are Not Automatically Eigenvectors

The defining equation for a generalized eigenvector is weaker than the eigenvector equation. For \(\mathbf{v}_2\) in a chain, \((T-\lambda\operatorname{id}_V)\mathbf{v}_2=\mathbf{v}_1\), and the right-hand side is nonzero. Thus \(\mathbf{v}_2\) is not an eigenvector. More generally, the chain equations guarantee that the first vector is an eigenvector; they do not guarantee that later vectors are.

Common wrong calculation: from \((\mathbf{A}-3\mathbf{I}_2)\mathbf{v}_2=\mathbf{v}_1\), one might incorrectly conclude that \(\mathbf{A}\mathbf{v}_2=3\mathbf{v}_2\). Adding \(3\mathbf{v}_2\) to both sides instead gives \(\mathbf{A}\mathbf{v}_2=3\mathbf{v}_2+\mathbf{v}_1\). Since \(\mathbf{v}_1\ne\mathbf{0}\), this is not the eigenvector equation.

What Geometric Multiplicity Does—and Does Not—Tell You

The geometric multiplicity of an eigenvalue \(\lambda\) is the dimension of its eigenspace \(E_{\lambda}=\ker(T-\lambda\operatorname{id}_V)\). For Jordan form, it has a precise interpretation: it counts the Jordan blocks for \(\lambda\), not the total number of vectors in those blocks and not the size of the largest block.

Proof. In a Jordan basis, the space is a direct sum of the subspaces belonging to the Jordan blocks. On a block of size \(k\) for \(\lambda\), subtracting \(\lambda\) times the identity leaves a matrix with ones on the superdiagonal and zeros elsewhere. If a vector in that block has coordinates \((x_1,\dots,x_k)^{\top}\), the equation for belonging to the kernel is \(x_2=0,\dots,x_k=0\); \(x_1\) is free. Thus that block contributes exactly one independent eigenvector, the first vector of its chain. Blocks for other eigenvalues contribute no vectors to this eigenspace: their block matrices after subtracting \(\lambda\) have nonzero diagonal \(\mu-\lambda\) and are invertible, since their triangular systems have a unique zero solution. The full eigenspace is the direct sum of the one-dimensional contributions from the \(\lambda\)-blocks, so its dimension is the number of those blocks. \(\square\)

Consequently, knowing the geometric multiplicity gives the number of chains, but not their lengths. The total dimension of the generalized eigenspace gives the sum of the lengths, and several different lists of lengths can have the same number of chains and the same sum.

A Short Diagnostic Routine

When checking a proposed Jordan calculation, use the defining equations before relying on a remembered pattern.

A frequent practical mistake is to inspect only the final matrix and assume that a familiar-looking block proves the chain is correct. A matrix pattern can be checked independently: multiply it by each proposed basis vector and compare the result with the chain equations. This catches both an order reversal and a misplaced off-diagonal entry. Likewise, a count of eigenvectors can check the number of blocks, but it cannot replace checks on the chain lengths.