Abstract
“Is it diagonalisable” is a question answered yes or no. This paper writes in numbers what that answer fails to guarantee. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, the law is $\varepsilon^{1/n}$. Putting $10^{-16}$ in the corner of an $n\times n$ Jordan block gives eigenvalues of size $10^{-4}$ at $n=4$, $0.0251188643151$ at $n=10$ and $\mathbf{0.158489319247}$ at $\mathbf{n=20}$ — with a ratio to the prediction of $1.0000000000$ in every case (Section 2).
Second, the eigenvalues lie evenly around a circle. At $n=12$ every radius is $0.0464158883361$ and every angular gap is $0.52359878$, which is $2\pi/12$ (Section 3).
Third, this is the core. Under a random disturbance of the same size, the Jordan block moves by $\mathbf{0.14973076}$ at $n=20$ while a well-separated diagonal matrix moves by $6.22\times10^{-15}$ — a factor of $\mathbf{10^{15}}$ for the same disturbance (Section 4).
Fourth, the yes or no is in truth a quantity. A $2\times2$ matrix whose eigenvalues differ by $\delta$ is diagonalisable at every $\delta$, and the condition number of its eigenvectors reaches $\mathbf{2\times10^{9}}$ by $\delta=10^{-9}$ (Section 5).
Fifth, a Jordan block never appears in floating point. Adding a random disturbance of size $10^{-16}$ gave $n$ distinct eigenvalues in all two hundred trials, at $n=5$, $10$ and $20$ alike (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the Jordan form, Wilkinson's perturbation analysis of eigenvalues and pseudospectra are all standard. No numerical method is proposed — this paper measures with existing tools. The Schur decomposition is not computed — that a stable computation exists is cited and no more. The general theory of non-normal matrices is not developed — what appears is the Jordan block and one family of $2\times2$ matrices. Nothing is claimed about any application. Relation to Paper 219: there, a lens free of aberration existed at a single point in the space of materials — the Jordan block here is likewise a single point in the space of matrices, and moreover arithmetic can never stand on it. Relation to Paper 217: there, the truth of a theorem switched at $p=\infty$ while its force faded continuously — the same thing happens here. What is added is confirming the law at six dimensions with a ratio of $1.0000000000$, measuring that the eigenvalues are evenly spaced on a circle, comparing the movement under the same disturbance with a diagonal matrix and obtaining a factor of $10^{15}$, measuring the condition number and the movement against $\delta$, finding the crossing of the two regimes at $\varepsilon=\delta^2$, counting that a Jordan block appears in none of two hundred trials, and confirming that the radius of the pseudospectrum equals $\varepsilon^{1/n}$.
1. Notation and Position
An $n\times n$ Jordan block has ones just above the diagonal and zeros elsewhere. Its only eigenvalue is $0$, with multiplicity $n$, and it has only one eigenvector. Throughout, $\varepsilon=10^{-16}$, about the gap between $1$ and its neighbour in double precision.
Position: In Paper 217 the uniqueness of the midpoint switched at $p=\infty$ while the softness was gradual — in Paper 219 the perfect lens existed at a single point. This paper is where those two happen at once.
2. The Law: It Moves by the $n$-th Root of $\varepsilon$
Check 2.1: put $\varepsilon=10^{-16}$ in the bottom-left corner of the block.
| $n$ | Largest $|$eigenvalue$|$ | $\varepsilon^{1/n}$ | Ratio |
|---|---|---|---|
| $2$ | $10^{-8}$ | $10^{-8}$ | $1.0000000000$ |
| $4$ | $10^{-4}$ | $10^{-4}$ | $1.0000000000$ |
| $8$ | $0.01$ | $0.01$ | $1.0000000000$ |
| $10$ | $0.0251188643151$ | $0.0251188643151$ | $1.0000000000$ |
| $16$ | $0.1$ | $0.1$ | $1.0000000000$ |
| $20$ | $\mathbf{0.158489319247}$ | $0.158489319246$ | $1.0000000000$ |
Observation 2.2 Altering the sixteenth digit alone moves the eigenvalues of a twenty by twenty matrix by $\mathbf{0.16}$. That is $10^{15}$ times $\varepsilon$ itself — and the sentence “matrices that are close have close eigenvalues” does not hold here.
3. And They Scatter Evenly Around a Circle
Check 3.1: the radii and arguments at $n=12$.
| Measured | Value |
|---|---|
| Radii, least to greatest | $0.0464158883361$ to $0.0464158883361$ |
| Angular gaps, least and greatest | $0.52359878$ and $0.52359878$ |
| $2\pi/12$ | $0.52359878$ |
Observation 3.2 The twelve eigenvalues are the twelfth roots of $\varepsilon$, spaced perfectly evenly around a circle. One eigenvalue of multiplicity $n$ has turned into $n$ simple ones — and not one of them is real: the original matrix has the single eigenvalue $0$.
4. The Core: Under the Same Disturbance a Diagonalisable Matrix Moves $10^{15}$ Times Less
Check 4.1: twenty random disturbances of norm $\varepsilon$ applied to each.
| Matrix | $n$ | Largest movement | Movement $\div\ \varepsilon$ |
|---|---|---|---|
| Jordan block | $10$ | $0.02216382$ | $2.22\times10^{14}$ |
| Diagonal, well separated | $10$ | $4.44\times10^{-15}$ | $44.4$ |
| Jordan block | $20$ | $\mathbf{0.14973076}$ | $\mathbf{1.50\times10^{15}}$ |
| Diagonal, well separated | $20$ | $\mathbf{6.22\times10^{-15}}$ | $\mathbf{62.2}$ |
($\varepsilon^{1/n}$ at $n=20$ is $0.15848932$.)
Observation 4.2 Under disturbances of exactly the same size, one moves by $6\times10^{-15}$ and the other by $0.15$ — a factor of $\mathbf{10^{15}}$. The size of the matrix and the size of the disturbance are the same in both; the only difference is whether the eigenvectors form a basis.
5. The Yes or No Is in Truth a Quantity
Let $A(\delta)$ be the $2\times2$ matrix whose $(1,2)$ entry is $1$ and whose $(2,2)$ entry is $\delta$, the other two being $0$. Its eigenvalues are $0$ and $\delta$, so it is diagonalisable for every $\delta\neq0$.
Check 5.1:
| $\delta$ | Condition number of the eigenvectors | Movement under $\varepsilon$ |
|---|---|---|
| $10^{-1}$ | $20.0499$ | $9.99\times10^{-16}$ |
| $10^{-3}$ | $2000$ | $10^{-13}$ |
| $10^{-5}$ | $200000$ | $10^{-11}$ |
| $10^{-7}$ | $2\times10^{7}$ | $9.90\times10^{-10}$ |
| $10^{-9}$ | $\mathbf{2\times10^{9}}$ | $\mathbf{9.51\times10^{-9}}$ |
Observation 5.2 “Diagonalisable” is true at every $\delta$, and the basis becomes arbitrarily bad. The word is yes or no and what it stands for is continuous — what switches at $\delta=0$ is only the word, and nothing switches in the numbers.
6. Two Regimes, and Where They Cross
Check 6.1: for the same $A(\delta)$, vary the disturbance.
| $\delta$ | $\varepsilon$ | Movement | $\varepsilon/\delta$ | $\sqrt{\varepsilon}$ |
|---|---|---|---|---|
| $10^{-3}$ | $10^{-14}$ | $10^{-11}$ | $10^{-11}$ | $10^{-7}$ |
| $10^{-3}$ | $10^{-8}$ | $9.90\times10^{-6}$ | $10^{-5}$ | $10^{-4}$ |
| $10^{-3}$ | $10^{-6}$ | $6.18\times10^{-4}$ | $10^{-3}$ | $10^{-3}$ |
| $10^{-5}$ | $10^{-14}$ | $10^{-9}$ | $10^{-9}$ | $10^{-7}$ |
| $10^{-5}$ | $10^{-10}$ | $6.18\times10^{-6}$ | $10^{-5}$ | $10^{-5}$ |
Observation 6.2 While the disturbance is small the movement follows $\varepsilon/\delta$, and once $\varepsilon$ passes $\delta^2$ it changes over to $\sqrt{\varepsilon}$. The crossing is at $\varepsilon=\delta^2$ — and the change-over is smooth: there is no moment at which the matrix stops being diagonalisable. It has the shape of what happened in Paper 217 as $p$ was raised.
7. A Jordan Block Never Appears in Floating Point
Check 7.1: add a random matrix of size $\varepsilon$ to the block and count the distinct eigenvalues, two hundred times each.
| $n$ | Distinct eigenvalues | Out of |
|---|---|---|
| $5$ | always $5$ | $200/200$ |
| $10$ | always $10$ | $200/200$ |
| $20$ | always $20$ | $200/200$ |
Observation 7.2 Not once does a repetition survive. Matrices with a repeated eigenvalue form a set of measure zero, and rounding always pushes one outside it — so “this matrix is a Jordan block” can never be confirmed by computation. As in Paper 219 the perfect lens stood at a single point in the space of materials, so here a single point in the space of matrices, and arithmetic cannot stand on it.
8. What Is Honest Instead: The Pseudospectrum, and the Separator and the Fence
Instead of the eigenvalues, take the set of $z$ where the smallest singular value of $zI-J$ falls below $\varepsilon$: the pseudospectrum.
Check 8.1: its radius.
| $n$ | Radius of the pseudospectrum | $\varepsilon^{1/n}$ | Ratio |
|---|---|---|---|
| $4$ | $0.0001000000003$ | $0.0001$ | $1.000000$ |
| $8$ | $0.01000012501$ | $0.01$ | $1.000013$ |
| $12$ | $0.04642423441$ | $0.04641588834$ | $1.000180$ |
| $16$ | $0.1000629138$ | $0.1$ | $1.000629$ |
Observation 8.2 The radius of the pseudospectrum is the radius to which the eigenvalues actually moved. The spectrum says “one point” and the pseudospectrum says “a disc” — and what arithmetic sees is the disc.
Observation 8.3 (the separator) What separates the senses of “eigenvalue” is which matrix is being spoken of — that matrix exactly, or everything near it. The first gives the single point $0$ and the second a disc of radius $\varepsilon^{1/n}$, and both are correct.
Observation 8.4 (the fence) The fence around “matrices that are close have close eigenvalues” is that the eigenvectors form a basis. Outside it the sentence is false, and false by a factor of $10^{15}$. What happened in Paper 220 happens here: the test “is it diagonalisable” is passed, and passing it guarantees nothing — what would guarantee something is the condition number, which appears nowhere in the answer.
9. The Census and the Close
What this paper measured:
- The law is $\varepsilon^{1/n}$ (Check 2.1): a ratio of $1.0000000000$ at all six dimensions.
- Evenly spaced on a circle (Check 3.1): $2\pi/12=0.52359878$.
- A factor of $10^{15}$ under the same disturbance (Check 4.1): $0.14973076$ against $6.22\times10^{-15}$.
- The yes or no is a quantity (Check 5.1): a condition number of $2\times10^{9}$.
- The crossing is at $\varepsilon=\delta^2$ (Check 6.1): and the change-over is smooth.
- The Jordan block never appears (Check 7.1): $n$ distinct in $200/200$.
- The pseudospectrum gives the true radius (Check 8.1): six digits at $n=4$.
References
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Reference
Yamagishi, Y. (2026). Change the Last Bit and an Eigenvalue Moves by 0.16 [Paper 221]. YOI: https://www.yamagishiyuuki.com/yoi/221
DOI: 10.5281/zenodo.22120012
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 221. This paper is a preprint and has not been peer reviewed.