Jordan Data and Consistency Checks

Several Descriptions of the Same Jordan Blocks

A Jordan form can be described in several ways: by the characteristic polynomial, by eigenspace dimensions, by the growth of kernels of powers, or by the minimal polynomial. Each description emphasizes a different feature of the blocks, but the data cannot vary independently. Connecting them gives a practical way to reconstruct block sizes and to detect claims that cannot all be true.

Work over \(\mathbb{C}\), and let \(\mathbf{A}\in\mathbb{C}^{n\times n}\). By the Jordan Form Existence Theorem, \(\mathbf{A}\) is similar to a block diagonal matrix of Jordan blocks. Similar matrices have the same characteristic and minimal polynomials, and corresponding eigenspaces and kernels have the same dimensions. Thus we may read the data from the Jordan form. For an eigenvalue \(\lambda\), write a block of size \(s\) as \(\mathbf{J}_s(\lambda)=\lambda\mathbf{I}_s+\mathbf{N}_s\), where \(\mathbf{N}_s\) has \(1\)s immediately above the diagonal and \(0\)s elsewhere.

Let the sizes of the Jordan blocks for \(\lambda\) be \(s_1,\dots,s_r\), counting repeated sizes as separate blocks. The key quantities will be the total size \(\sum_i s_i\), the number \(r\) of blocks, and the largest size. The next results establish how each of the data above records one of these quantities.

What the Characteristic Polynomial and Eigenspace Count

Proof. In the Jordan form, the determinant of each block \(\mathbf{J}_{s_i}(\lambda)-z\mathbf{I}_{s_i}\) is \((\lambda-z)^{s_i}\), since the block is triangular and all its diagonal entries are \(\lambda-z\). Blocks for eigenvalues other than \(\lambda\) contribute factors that are nonzero at \(z=\lambda\). Therefore the multiplicity of the root \(\lambda\) in \(\det(\mathbf{A}-z\mathbf{I})\) is \(\sum_i s_i\). On a block for \(\lambda\), the equation \((\mathbf{J}_{s_i}(\lambda)-\lambda\mathbf{I})\mathbf{x}=\mathbf{0}\) is \(\mathbf{N}_{s_i}\mathbf{x}=\mathbf{0}\). Because \(\mathbf{N}_{s_i}(x_1,\dots,x_{s_i})^\top=(x_2,\dots,x_{s_i},0)^\top\), its kernel consists precisely of vectors with \(x_2=\cdots=x_{s_i}=0\), so that block contributes one dimension. On a block for a different eigenvalue \(\mu\), \(\mathbf{J}_{s_i}(\mu)-\lambda\mathbf{I}\) is triangular with nonzero diagonal \(\mu-\lambda\), hence has zero kernel. The full eigenspace dimension is consequently one for each block for \(\lambda\) and zero for every other block. Thus \(\dim E_\lambda=r\). \(\square\)

So the characteristic polynomial gives the sum of the block sizes, while the eigenspace dimension gives the number of blocks. Neither quantity alone usually determines every size. The kernel dimensions give more detailed information: the increase from one power to the next counts how many blocks are still large enough to contribute another dimension.

Kernel Growth Records Block Sizes

Proof. On a block of size \(s\) for \(\lambda\), the relevant power is \(\mathbf{N}_s^k\). If \(k<s\), then \(\mathbf{N}_s^k(x_1,\dots,x_s)^\top\) has first \(s-k\) entries \(x_{k+1},\dots,x_s\) and its remaining entries are zero. Its kernel therefore has dimension \(k\): precisely \(s-k\) independent coordinates are forced to vanish, leaving \(k\) free coordinates. If \(k\geq s\), the power is the zero matrix and its kernel has dimension \(s\). Thus this block contributes \(\min(k,s)\) to \(d_k(\lambda)\). Adding the contributions from all blocks gives \(d_k(\lambda)=\sum_{i=1}^r\min(k,s_i)\). For each block of size \(s_i\), the difference \(\min(k,s_i)-\min(k-1,s_i)\) is \(1\) when \(s_i\geq k\), and \(0\) when \(s_i<k\). Summing these differences shows that \(b_k\) counts blocks of size at least \(k\). Subtracting the number of blocks of size at least \(k+1\) from the number of size at least \(k\) counts exactly the blocks of size \(k\), namely \(b_k-b_{k+1}\). Finally, \(b_k>0\) exactly while at least one block has size at least \(k\), so the last such \(k\) is the largest block size. \(\square\)

This theorem gives useful consistency rules. The increments \(b_k\) must be nonnegative integers and cannot increase as \(k\) increases. They eventually become zero. Their sum is the algebraic multiplicity, because each block of size \(s\) contributes \(1\) to \(b_1,\dots,b_s\), for a total contribution of \(s\). Also, \(d_k(\lambda)\) cannot exceed the algebraic multiplicity and becomes constant once \(k\) reaches the largest block size.

The Minimal Polynomial Records the Largest Block

Proof. Let \(L\) be the largest block size for \(\lambda\). First consider any polynomial \(q(x)\), and write \(q(x)=(x-\lambda)^a g(x)\), where \(a\geq0\) and \(g(\lambda)\ne0\). On a block \(\mathbf{J}_s(\lambda)=\lambda\mathbf{I}_s+\mathbf{N}_s\), we have \(q(\mathbf{J}_s(\lambda))=\mathbf{N}_s^a g(\mathbf{J}_s(\lambda))\). The matrix \(g(\mathbf{J}_s(\lambda))\) is invertible: it has the form \(g(\lambda)\mathbf{I}_s+\mathbf{K}\), where \(\mathbf{K}\) is a polynomial in \(\mathbf{N}_s\) with zero constant term, so \(\mathbf{K}\) is nilpotent. Since \(g(\lambda)\ne0\), factoring out \(g(\lambda)\) leaves the identity plus a nilpotent matrix, whose inverse is a finite geometric sum. Also, \(g(\mathbf{J}_s(\lambda))\) commutes with \(\mathbf{N}_s\), since it is a polynomial in \(\mathbf{N}_s\). Hence \(q(\mathbf{J}_s(\lambda))=\mathbf{O}\) exactly when \(\mathbf{N}_s^a=\mathbf{O}\), which holds exactly when \(a\geq s\). Any polynomial that annihilates \(\mathbf{A}\) must annihilate every Jordan block, so its multiplicity at \(\lambda\) must be at least \(L\). Conversely, the product of \((x-\mu)^{L_\mu}\) over all eigenvalues \(\mu\), where \(L_\mu\) is the largest block size for \(\mu\), annihilates every block: on a block for \(\mu\), the factor \((x-\mu)^{L_\mu}\) evaluates to a zero matrix, and all polynomial factors commute. It therefore annihilates the Jordan form and, by similarity, \(\mathbf{A}\). The monic annihilating polynomial of least degree consequently has multiplicity exactly \(L\) at \(\lambda\). \(\square\)

For consistency checks, the minimal-polynomial exponent for \(\lambda\) must equal the last index \(k\) with \(b_k>0\). The exponent is not the number of blocks, nor is it the algebraic multiplicity unless there happens to be one block of that total size.

Rejecting Inconsistent Data

Common Pitfalls and a Practical Checklist

Do not mistake an increment for an exact block count. In the reconstruction example, \(b_2=3\) means there are three blocks of size at least \(2\), not three blocks of size exactly \(2\). The exact number of size-\(2\) blocks is \(b_2-b_3=3-2=1\). Treating the increment itself as the exact-size count would incorrectly report three size-\(2\) blocks and lose the information from the later increments.

Do not infer the minimal-polynomial exponent from the eigenspace dimension. The eigenspace dimension counts all blocks, whereas the exponent identifies the largest one. For example, block sizes \(4,2,1\) give eigenspace dimension \(3\) but minimal-polynomial exponent \(4\). Conversely, three blocks of size \(1\) have eigenspace dimension \(3\) and exponent \(1\).

When checking a proposed collection of Jordan data for each eigenvalue, use the following order:

1. Check that the algebraic multiplicity is the sum of the block sizes, or, when using kernel data, that \(\sum_{k\geq1}b_k\) equals that multiplicity.
2. Check that \(b_1=d_1\) equals the geometric multiplicity and that the increments are nonnegative, nonincreasing integers which eventually reach zero.
3. Recover the exact number of size-\(k\) blocks as \(b_k-b_{k+1}\); every such count must be a nonnegative integer.
4. Check that the largest \(k\) with \(b_k>0\) agrees with the minimal-polynomial exponent.
5. Across all eigenvalues, check that the algebraic multiplicities sum to \(n\), the size of the matrix.

These checks are necessary: if any one fails, the data cannot describe a Jordan form. When all the data are given, the exact-size counts determine the block sizes for each eigenvalue, so they provide a direct consistency test rather than merely a list of suggestive numerical summaries.