On the Growth Factor of Random Matrices

Abstract: Gaussian elimination is the oldest and most popular method for solving a linear system. Its numerical stability for a given matrix is controlled by the growth factor, a measure of how large entries can become during elimination. In this work, we prove a number of new results regarding the growth factor of random matrices. First, we provide tight estimates for the growth factor of a matrix preconditioned by a Haar orthogonal matrix without pivoting and characterize the asymptotic distribution of the growth factor of Gaussian matrices without pivoting. Second, and most notably, we prove that the growth factor of an $n \times n$ Gaussian matrix under partial pivoting is rarely much larger than $\sqrt{n}$, resolving an old conjecture of Nick Trefethen. The same techniques used to prove this conjecture also provide an improved smoothed analysis of the growth factor under partial pivoting.

• The paper proves that for an $n imes n$ Gaussian matrix $G$, the growth factor using Gaussian elimination with partial pivoting (GEPP) is asymptotically $\sqrt{n}\, \exp\big(C_{\alpha} \sqrt{\,\log n}\big)$ with high probability, resolving Trefethen’s conjecture.
• Without pivoting, the growth factor scales as $n^{3/2}$ and follows an inverse-exponential distribution, highlighting the significant role of partial pivoting in controlling growth.
• Truncated inverse control and spectral recursion are used to achieve a rigorous bound on the growth factor, while smoothed analysis explores the impact of perturbations on the growth factor.

Problem setting and principal claims

Gaussian elimination is numerically stable when intermediate quantities remain controlled. The relevant measure is the growth factor

$$\operatorname{growth}(A) = \max\left\{ \|L\|_{\max}, \frac{\|U\|_{\max}}{\|A\|_{\max}} \right\},$$

for an LU factorization, with $\|\cdot\|_{\max}$ denoting the entrywise maximum norm. Large growth factors amplify round-off error and increase the precision required for a reliable factorization.

The paper studies the asymptotic behavior of this quantity for several random-matrix models and pivoting strategies. Its central result concerns Gaussian elimination with partial pivoting (GEPP): for an $n\times n$ standard Gaussian matrix $G$, the growth factor is at most

$\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$

except with probability at most $n^{-\alpha}$, for every fixed $\alpha>0$. Equivalently,

$$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$

where $\Pi$ denotes the partial-pivoting row permutation. This proves Trefethen’s conjecture that the growth factor exceeds $n^{1/2+\alpha}$ with superpolynomially small probability for every $\|\cdot\|_{\max}$0 (2610.06785).

The result is notable because the deterministic worst-case growth factor under partial pivoting is $\|\cdot\|_{\max}$1, and this exponential upper bound is attained by explicit matrices. The paper therefore establishes a strong separation between worst-case and Gaussian average-case behavior. It also gives two complementary results: Haar orthogonal preconditioning without pivoting typically produces growth on the $\|\cdot\|_{\max}$2 scale, and Gaussian matrices without pivoting have an explicit inverse-exponential limiting law at precisely that scale.

Gaussian elimination and the role of partial pivoting

At elimination step $\|\cdot\|_{\max}$3, the Schur complement of the leading $\|\cdot\|_{\max}$4 block contains the remaining quantities generated by the factorization. Without pivoting, small leading principal minors can produce large entries in the Schur complements. Partial pivoting selects, at each stage, the row with largest magnitude in the current first column. In exact arithmetic, this guarantees

$\|\cdot\|_{\max}$5

so the principal difficulty is bounding the maximum entry of the evolving Schur complements, or equivalently the maximum entry of $\|\cdot\|_{\max}$6.

For a fixed stage $\|\cdot\|_{\max}$7, the paper encodes the elimination process through a matrix

$\|\cdot\|_{\max}$8

where $\|\cdot\|_{\max}$9 is the matrix formed by the first $n\times n$0 pivot rows and $n\times n$1 restores the appropriate row ordering. Conditional on the first $n\times n$2 revealed columns, every entry of the next Schur complement is Gaussian with variance given by the squared Euclidean norm of a row of $n\times n$3. Thus, the growth problem reduces to controlling these row norms uniformly over all stages.

This reduction exposes the principal technical obstacle. Direct control of $n\times n$4 is impossible at the required level because inverse Gaussian matrices have heavy tails, and their inverse Frobenius norms can have infinite expectation after normalization. The paper resolves this by discarding the largest $n\times n$5 singular values of $n\times n$6 and controlling the truncated quantity

$n\times n$7

where the sum excludes the $n\times n$8 largest singular values of the inverse, equivalently the $n\times n$9 smallest singular values of $G$0.

The proof then decomposes $G$1 into a controlled component and a rank-$G$2 component. The controlled component is governed by $G$3. The rank-$G$4 component is neutralized by exposing an additional $G$5 Gaussian columns. These columns provide enough independent randomness to regularize the exceptional directions, while the partial-pivoting multipliers contribute at most a factor $G$6 in the relevant row norms. Choosing

$G$7

balances the regularization cost against the singular-value estimates and produces the subpolynomial factor $G$8.

The main GEPP theorem

The paper proves that, for every $G$9, there are constants $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$0 and $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$1 such that

$\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$2

for all $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$3 (2610.06785).

This is stronger than a polynomial high-probability bound. It gives the sharp exponent $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$4 up to a factor that is $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$5. Since

$\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$6

the theorem directly implies Trefethen’s conjecture: $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$7 for any $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$8 and all sufficiently large $\sqrt n\,\exp\!\big(C_\alpha\sqrt{\log n}\big)$9.

The exponent $n^{-\alpha}$0 cannot be improved. The paper invokes a lower bound based on the determinant and the geometric mean of the singular values: with probability tending to one,

$n^{-\alpha}$1

while $n^{-\alpha}$2. Consequently, the growth factor is at least $n^{-\alpha}$3 with high probability. The upper and lower bounds therefore identify the correct logarithmic scale, not merely an upper estimate.

The theorem does not identify the limiting distribution of $n^{-\alpha}$4. Numerical experiments suggest that the normalized fluctuations may be approximately Gumbel after centering and scaling, and that the uncentered distribution may narrow as $n^{-\alpha}$5 increases. These observations remain conjectural; no concentration around a deterministic constant or limiting extreme-value law is established.

Figure 1: Empirical densities and fitted Gumbel-type normalizations for $n^{-\alpha}$6.

Truncated inverse control and spectral recursion

The central spectral result is a uniform bound on truncated inverse norms of the pivot blocks. For every $n^{-\alpha}$7, with probability at least $n^{-\alpha}$8,

$n^{-\alpha}$9

simultaneously for all $\alpha>0$0 (2610.06785).

The proof constructs a recursive sequence of singular-value scales. Starting from a deficiency parameter $\alpha>0$1, the recursion grows approximately according to

$\alpha>0$2

At each extension by $\alpha>0$3 columns, a lower bound on the relevant singular value deteriorates by an explicitly tracked exponential factor. The recursion reaches the ambient dimension in only $\alpha>0$4 stages, explaining the $\alpha>0$5 term in the final estimate.

The extension argument combines several ingredients:

• concentration for Gaussian matrix products;
• bounds on pseudoinverses of wide Gaussian matrices;
• lower-tail estimates for small singular values of Gaussian matrices;
• a singular-value inequality for appending rows;
• a conditioning argument that preserves the partial-pivoting structure.

The resulting estimate is weaker than the heuristic bound $\alpha>0$6 by a subpolynomial factor, but it is sufficiently strong for the growth analysis. The implication is that the potentially pathological small singular directions are sparse enough to be regularized by a logarithmically small number of additional Gaussian columns.

Gaussian matrices without pivoting

The behavior without pivoting is substantially worse. Let $\alpha>0$7 be the LU factorization of a standard Gaussian matrix without row exchanges. The paper proves the distributional convergence

$\alpha>0$8

where $\alpha>0$9 has inverse-exponential distribution

$$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$0

Thus, unlike GEPP, no-pivot Gaussian elimination has a genuinely heavy-tailed $$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$1-scale growth factor (2610.06785).

The argument isolates the dominant contribution from $$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$2. If

$$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$3

then

$$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$4

and the variables $$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$5 are independent. Their maximum satisfies an inverse-exponential extreme-value limit after normalization by

$$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$6

The additional Gaussian maximum arising within each row of $$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$7 contributes a factor asymptotic to $$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$8. Since $$\frac{\log \operatorname{growth}(\Pi(G))}{\log n} \overset{p}{\longrightarrow} \frac12,$$9 is asymptotic to $\Pi$0, these logarithmic factors cancel in the normalized growth factor, leaving the inverse-exponential limit above. The lower-triangular factor is negligible on the $\Pi$1 scale.

The convergence is numerically rapid. For $\Pi$2, one million samples yield a Kolmogorov–Smirnov distance of approximately $\Pi$3 from the limiting inverse-exponential distribution. This numerical agreement supports the exact asymptotic description, while also emphasizing the qualitative difference between no pivoting and partial pivoting.

Figure 2: Empirical convergence of $\Pi$4 without pivoting to the inverse-exponential limiting distribution.

The contrast between the two procedures is sharp. Without pivoting, the normalized growth is $\Pi$5 with polynomially small probability because of the inverse-exponential tail. Under partial pivoting, the corresponding normalized quantity is $\Pi$6 with overwhelming probability. Partial pivoting therefore changes not only constants or tail exponents, but the polynomial scale of the random growth factor.

Haar orthogonal preconditioning

The paper also analyzes elimination without pivoting after left multiplication by a Haar-distributed orthogonal matrix. For every fixed nonsingular $\Pi$7 and $\Pi$8 sampled uniformly from $\Pi$9,

$n^{1/2+\alpha}$0

The proof uses a distributional description of inverses of square submatrices of a Haar orthogonal matrix. For a suitable submatrix $n^{1/2+\alpha}$1 and fixed unit vectors, the relevant bilinear forms are controlled through the inverse of a single coordinate of a uniformly distributed spherical vector. A union bound over all elimination stages yields the $n^{1/2+\alpha}$2 estimate.

The paper further proves that the exponent $n^{1/2+\alpha}$3 is sharp for nonadaptive orthogonal preconditioning. A Gaussian matrix can be viewed as having orthogonally invariant row and column spaces, and the no-pivot result shows that Gaussian growth itself is asymptotically $n^{1/2+\alpha}$4. Consequently, a random orthogonal preconditioner does not reproduce the $n^{1/2+\alpha}$5 behavior of partial pivoting. This is a substantive negative comparison: randomization through a fixed orthogonal transformation is insufficient to match the adaptive row selection provided by GEPP.

The tail behavior is also materially weaker. The Haar-preconditioned bound has inverse-polynomial dependence on the failure probability $n^{1/2+\alpha}$6, whereas the GEPP Gaussian theorem permits arbitrarily high polynomial confidence. Thus, both the typical scale and the available tail control favor adaptive partial pivoting over nonadaptive orthogonal preconditioning in this model.

Smoothed analysis

The second major extension treats matrices of the form

$n^{1/2+\alpha}$7

For every $n^{1/2+\alpha}$8, the paper proves

$n^{1/2+\alpha}$9

for sufficiently large $\|\cdot\|_{\max}$00 (2610.06785).

This improves the classical smoothed-analysis estimate of the form

$\|\cdot\|_{\max}$01

to a bound with only a $\|\cdot\|_{\max}$02 exponent on $\|\cdot\|_{\max}$03. In particular, polynomially small perturbations yield growth bounded by $\|\cdot\|_{\max}$04 with overwhelming probability. The paper explicitly states that the likely optimal dependence is polynomial in both $\|\cdot\|_{\max}$05 and $\|\cdot\|_{\max}$06, but does not establish that stronger claim.

The general smoothed theorem tracks the mean matrix through

$\|\cdot\|_{\max}$07

and introduces a product penalty

$\|\cdot\|_{\max}$08

which records the accumulated cost of nonzero means along the singular-value recursion. The resulting bound has the structure

$\|\cdot\|_{\max}$09

The proof requires a careful distinction between Gaussian randomness and the deterministic mean. Conditional nonpivot rows are Gaussian vectors restricted to symmetric convex sets induced by the pivot constraints. A log-concavity argument shows that these conditioned vectors retain subgaussian concentration, with the deterministic mean contributing explicit norm-dependent losses. This is where the smoothed theorem differs technically from the centered Gaussian case; the core low-rank regularization mechanism remains the same.

Figure 3: Truncated inverse norms for pivoted and unpivoted leading blocks, illustrating why removing the smallest singular directions yields substantially stronger control.

Limitations and open questions

The paper establishes the sharp polynomial exponent for Gaussian GEPP but does not determine the asymptotic constant or limiting distribution of

$\|\cdot\|_{\max}$10

The numerical evidence for Gumbel-type fluctuations is suggestive but not a theorem, and the apparent narrowing of the empirical densities leaves open whether the normalized growth converges to a deterministic constant.

The smoothed bound is also not believed to be optimal. Its dependence on $\|\cdot\|_{\max}$11 is $\|\cdot\|_{\max}$12, whereas the paper indicates that an $\|\cdot\|_{\max}$13-type bound is likely. Establishing polynomial dependence would require eliminating or substantially reducing the losses accumulated in the recursive singular-value extension argument.

A further limitation is model specificity. The principal theorem concerns real standard Gaussian matrices. The proof relies heavily on Gaussian conditioning, rotational invariance, small-ball estimates, and concentration for Gaussian matrix products. Extending the sharp $\|\cdot\|_{\max}$14 result to broader subgaussian, dependent, or discrete ensembles is not obtained here. In particular, the paper does not establish whether the same exponent and confidence bounds hold under general independent entries with only finite moments.

Finally, the analysis demonstrates that partial pivoting does not appear to worsen truncated inverse behavior relative to unpivoted Gaussian blocks, but it does not prove a monotonic comparison of these quantities. Such a comparison would provide a stronger structural explanation for why the greedy pivot rule achieves its observed average-case performance.

Conclusion

The paper gives a rigorous average-case theory for growth factors in Gaussian elimination. It proves that GEPP on an $\|\cdot\|_{\max}$15 Gaussian matrix has growth

$\|\cdot\|_{\max}$16

with overwhelming probability, resolving Trefethen’s conjecture and identifying the correct polynomial exponent. The proof combines truncated inverse control, recursive singular-value estimates, and a low-rank regularization argument based on additional Gaussian columns.

The comparison results clarify the mechanism behind this behavior. Gaussian elimination without pivoting, including Haar orthogonally preconditioned elimination, operates on the $\|\cdot\|_{\max}$17 scale and exhibits heavy inverse-exponential tails. Partial pivoting reduces the scale to $\|\cdot\|_{\max}$18 up to a subpolynomial factor. The smoothed analysis extends this mechanism beyond centered Gaussian matrices and improves the dependence on perturbation size from exponential-in-$\|\cdot\|_{\max}$19 exponents to $\|\cdot\|_{\max}$20 exponents, while leaving the optimal smoothed bound unresolved.