Jordan Form with Multiple Eigenvalues

One Eigenvalue at a Time

When a matrix has several distinct eigenvalues, its Jordan form is built by treating each eigenvalue separately. The vectors associated with one eigenvalue form a generalized eigenspace, an invariant subspace on which the matrix differs from a scalar multiple of the identity by a nilpotent map. Jordan chains can be found inside each such subspace; their lengths determine the Jordan blocks for that eigenvalue. The final Jordan matrix places all these local blocks together.

We work over \(\mathbb{C}\), so every eigenvalue of the matrix is included. If a matrix has real entries but nonreal eigenvalues, its Jordan form in this tutorial is understood over \(\mathbb{C}\). Let \(\mathbf{A}\in\mathbb{C}^{n\times n}\), and let \(\lambda\) be an eigenvalue. The generalized eigenspace for \(\lambda\) is the set of vectors killed by a sufficiently high power of \(\mathbf{A}-\lambda\mathbf{I}_n\). Using the exponent \(n\) is always sufficient.

On \(G_{\lambda}\), the map \(\mathbf{A}-\lambda\mathbf{I}_n\) is nilpotent, meaning that some power of it is the zero map. This is why the repeated-eigenvalue case applies to each summand. The key separation result is the Primary Decomposition Theorem: if the distinct eigenvalues of \(\mathbf{A}\) are \(\lambda_1,\dots,\lambda_r\), then \(\mathbb{C}^n=G_{\lambda_1}\oplus\cdots\oplus G_{\lambda_r}\). Here a direct sum means every vector has a unique expression as a sum of vectors from the listed subspaces. This standard result was established in the earlier material on primary decomposition.

Assembling the Local Jordan Forms

For a single eigenvalue \(\lambda_i\), the restriction of \(\mathbf{A}-\lambda_i\mathbf{I}_n\) to \(G_{\lambda_i}\) is nilpotent. The Jordan Form Existence Theorem for nilpotent maps, established earlier, supplies a basis of Jordan chains for this restriction. Adding \(\lambda_i\) to the diagonal of each nilpotent block gives Jordan blocks with eigenvalue \(\lambda_i\). The following result explains why doing this separately gives the Jordan form of the whole matrix.

Proof. By the Primary Decomposition Theorem, each vector of \(\mathbb{C}^n\) has a unique expression as a sum of vectors from \(G_{\lambda_1},\dots,G_{\lambda_r}\). A basis of each summand therefore combines to form a basis of the whole space: it spans by existence of the expressions, and is independent because any linear combination equal to \(\mathbf{0}\) gives a sum of vectors from the separate summands equal to \(\mathbf{0}\), so uniqueness forces each summand’s combination to be \(\mathbf{0}\). In the chosen basis, vectors from a chain in \(G_{\lambda_i}\) are mapped to linear combinations of vectors in that same chain. Thus no basis vector from one generalized eigenspace maps to a vector in another. The matrix is block diagonal across the summands, and within each summand its blocks are the Jordan blocks supplied by the chain basis. \(\square

This theorem gives a useful practical rule: do not try to build chains that mix eigenvalues. First identify each generalized eigenspace; then find its block sizes and chains; finally put the resulting blocks in any order. The order of the blocks does not change the similarity class, provided the basis vectors are ordered to match the chosen block order.

Finding Block Sizes Within a Summand

Fix an eigenvalue \(\lambda\), and write \(N=(\mathbf{A}-\lambda\mathbf{I}_n)|_{G_{\lambda}}\), the restriction to \(G_{\lambda}\). Define \(d_k=\dim\ker(N^k)\) for \(k\geq 0\), taking \(d_0=0\). The successive differences reveal how many blocks are present. A block of size \(s\) contributes \(\min(k,s)\) to \(d_k\): in a nilpotent Jordan block, the \(k\)th power kills the first \(\min(k,s)\) chain vectors and no others. Summing over the blocks gives the count formulas below.

Proof. Consider one nilpotent Jordan block of size \(s\), with chain basis \(\mathbf{v}_1,\dots,\mathbf{v}_s\) satisfying \(N\mathbf{v}_1=\mathbf{0}\) and \(N\mathbf{v}_j=\mathbf{v}_{j-1}\) for \(j\geq2\). Then \(N^k\mathbf{v}_j=\mathbf{0}\) exactly when \(j\leq k\), so the kernel on this block has dimension \(\min(k,s)\). Its contribution to \(d_k-d_{k-1}\) is \(1\) if \(s\geq k\), and \(0\) if \(s<k\). Adding these contributions over the blocks proves the first formula. The blocks of size exactly \(k\) are those of size at least \(k\) minus those of size at least \(k+1\). Substituting the first formula gives \((d_k-d_{k-1})-(d_{k+1}-d_k)=2d_k-d_{k-1}-d_{k+1}\). \(\square

A Common Mistake: Using Only Eigenvectors

A generalized eigenspace is not necessarily the ordinary eigenspace. In the first example, a tempting but incorrect calculation is to set \(G_2=\ker(\mathbf{A}-2\mathbf{I}_3)\). That kernel contains \(\mathbf{p}_1\), but it does not contain \(\mathbf{p}_2\), since \((\mathbf{A}-2\mathbf{I}_3)\mathbf{p}_2=\mathbf{p}_1\ne\mathbf{0}\). The mistaken conclusion would be that the eigenvalue \(2\) contributes only a size-\(1\) block. The correction is to use the generalized eigenspace: \((\mathbf{A}-2\mathbf{I}_3)^2\mathbf{p}_2=\mathbf{0}\), so \(\mathbf{p}_2\in G_2\), and the chain \(\mathbf{p}_2\mapsto\mathbf{p}_1\mapsto\mathbf{0}\) gives a size-\(2\) block.

Another useful check is that powers for one eigenvalue must remove the other generalized eigenspaces. In the first example, \((\mathbf{A}-2\mathbf{I}_3)\mathbf{p}_3=-3\mathbf{p}_3\), so \((\mathbf{A}-2\mathbf{I}_3)^k\mathbf{p}_3=(-3)^k\mathbf{p}_3\ne\mathbf{0}\) for every \(k\). The \(-1\) component therefore does not belong to \(G_2\). Once each summand has been isolated, kernel growth is measured on that summand, not by mixing vectors from different eigenvalues.

In practice, the work has three checks: the generalized eigenspaces must account for the full dimension; the Jordan block sizes on each summand must add to its dimension; and the concatenated chain vectors must form a basis in which the matrix acts as claimed. These checks catch both missing chains and incorrect assignments of blocks to eigenvalues.