Jordan Form for a Repeated Eigenvalue

From One Repeated Eigenvalue to Jordan Blocks

When a matrix has just one eigenvalue, its Jordan form is built from blocks for that eigenvalue. The remaining work is to determine the sizes of those blocks and find a basis that realizes them. Successive powers of \(\mathbf{A}-\lambda\mathbf{I}\) provide the block-size information; generalized eigenvector chains provide the basis. These are two parts of one calculation: the kernel dimensions tell us what chains to seek, and a final matrix multiplication verifies that the chains give the claimed Jordan form.

In this tutorial, let \(\mathbf{A}\in\mathbb{R}^{n\times n}\) have a single eigenvalue \(\lambda\in\mathbb{R}\), repeated with algebraic multiplicity \(n\). Set \(\mathbf{K}=\mathbf{A}-\lambda\mathbf{I}_n\). The earlier Jordan form existence result guarantees that such a matrix is similar over \(\mathbb{R}\) to a direct sum of Jordan blocks for \(\lambda\). We focus on computing that form and its basis. For an integer \(s\ge1\), the Jordan block \(J_s(\lambda)\) has \(\lambda\) on the diagonal, \(1\) on the superdiagonal, and \(0\) elsewhere.

The first vector in a chain is an eigenvector. The later vectors are generalized eigenvectors: they may not be eigenvectors themselves, but repeated application of \(\mathbf{K}\) eventually sends them to zero. The order matters. With the chain ordered as \((\mathbf{v}_1,\dots,\mathbf{v}_s)\), the matrix of \(\mathbf{A}\) on those basis vectors is \(J_s(\lambda)\).

Reading Block Sizes from Kernel Growth

For \(k\ge0\), write \(d_k=\dim\ker(\mathbf{K}^k)\), with \(\mathbf{K}^0=\mathbf{I}_n\), so \(d_0=0\). Since the only eigenvalue is \(\lambda\), the Jordan form of \(\mathbf{K}\) is a direct sum of nilpotent Jordan blocks. On a nilpotent block of size \(s\), the kernel of the \(k\)th power has dimension \(\min(k,s)\): it is spanned by the first \(\min(k,s)\) standard basis vectors. Thus each increase \(d_k-d_{k-1}\) counts the blocks of size at least \(k\).

Proof. Let the Jordan block sizes of \(\mathbf{K}\) be \(s_1,\dots,s_q\), as supplied by the Jordan form existence result. On a block of size \(s_i\), the kernel of the \(k\)th power has dimension \(\min(k,s_i)\), because the \(k\)th power maps each standard basis vector \(\mathbf{e}_j\) to zero exactly when \(j\le k\), or \(j\le s_i\) if \(k\ge s_i\). Kernels of a block diagonal matrix are direct sums of the block kernels, so \(d_k=\sum_{i=1}^q\min(k,s_i)\). For each block, \(\min(k,s_i)-\min(k-1,s_i)\) equals \(1\) if \(s_i\ge k\), and \(0\) otherwise. Summing gives the number of blocks of size at least \(k\). Subtracting the number of blocks of size at least \(k+1\) from the number of blocks of size at least \(k\) leaves exactly the blocks of size \(k\). \(\square\)

This rule also explains how far to compute. Once \(\mathbf{K}^k=\mathbf{O}\), the kernels have stabilized at all of \(\mathbb{R}^n\), and no block is longer than \(k\). In practice, compute \(\mathbf{K},\mathbf{K}^2,\dots\), find each kernel by solving a homogeneous system, and record its dimension. The sequence of increases determines the block sizes before any chains are chosen.

Turning Chains into a Similarity Basis

Proof. Consider one chain \((\mathbf{v}_1,\dots,\mathbf{v}_s)\). Its defining equations give \(\mathbf{A}\mathbf{v}_1=\lambda\mathbf{v}_1\) and \(\mathbf{A}\mathbf{v}_j=\mathbf{v}_{j-1}+\lambda\mathbf{v}_j\) for every \(j\ge2\). Therefore, in the ordered chain coordinates, the first column of the matrix is \((\lambda,0,\dots,0)^\top\), and column \(j\) has \(1\) in row \(j-1\), \(\lambda\) in row \(j\), and zero elsewhere. This is exactly \(J_s(\lambda)\). For multiple chains, each chain's images are combinations only of vectors in that same chain, so the matrix has these blocks on its diagonal and zeros between them. Because \(\mathbf{P}\) is the matrix whose columns are this basis, the coordinate matrix is \(\mathbf{P}^{-1}\mathbf{A}\mathbf{P}\). Hence it is the claimed block diagonal matrix. \(\square\)

The criterion gives a practical verification that does not require finding \(\mathbf{P}^{-1}\): check \(\mathbf{A}\mathbf{P}=\mathbf{P}\mathbf{J}\) column by column. If \(\mathbf{P}\) is invertible, this equation is equivalent to \(\mathbf{P}^{-1}\mathbf{A}\mathbf{P}=\mathbf{J}\). The chain equations verify the columns, while the fact that the columns form a basis verifies invertibility.

A Repeated Pair of Blocks

A Reliable Calculation and a Common Pitfall

For a matrix with one repeated eigenvalue, keep the block-count and basis-construction stages distinct. First, use the dimensions of \(\ker(\mathbf{K}^k)\) to determine how many chains of each length are needed. Next, find vectors satisfying the chain equations and arrange the chains in order. Finally, verify independence of all chosen vectors and check \(\mathbf{A}\mathbf{P}=\mathbf{P}\mathbf{J}\). A sequence of kernel dimensions identifies the sizes, but it does not by itself supply the actual columns of \(\mathbf{P}\).

Common pitfall: reversing a chain. In the second worked example, \(\mathbf{K}\mathbf{v}_2=\mathbf{v}_1\). If the columns are mistakenly ordered as \((\mathbf{v}_2,\mathbf{v}_1)\), then \(\mathbf{A}\mathbf{v}_2=\lambda\mathbf{v}_2+\mathbf{v}_1\) places the \(1\) below the diagonal in the first column. That is not the standard upper Jordan block \(J_2(\lambda)=\begin{bmatrix}\lambda&1\\0&\lambda\end{bmatrix}\). The correction is to order the chain as \((\mathbf{v}_1,\mathbf{v}_2)\): then the second column records \(\mathbf{A}\mathbf{v}_2=\mathbf{v}_1+\lambda\mathbf{v}_2\), placing the \(1\) in the superdiagonal position.

Another useful check is that the sum of all block sizes must equal \(n\). Each chain contributes one basis vector for every position in its block. If the proposed sizes do not sum to \(n\), the kernel counts have been misread or a block has been omitted. If they do sum to \(n\), the chain vectors still need to be checked for independence; the similarity equation alone does not make a singular \(\mathbf{P}\) into a change of basis.