Abstract
This paper establishes an almost sure limit for the operator norm of rectangular random matrices: Suppose $\\{v_{ij}\\}i = 1,2, \\cdots, j = 1,2, \\cdots$ are zero mean i.i.d. random variables satisfying the moment condition $E|\\nu_{11}|^n \\leqslant n^{\\alpha n}$ for all $n \\geqslant 2$, and some $\\alpha$. Let $\\sigma^2 = Ev^2_{11}$ and let $V_{pn}$ be the $p \\times n$ matrix $\\{v_{ij}\\}_{1\\leqslant i\\leqslant p; 1\\leqslant j\\leqslant n}$. If $p_n$ is a sequence of integers such that $p_n/n \\rightarrow y$ as $n \\rightarrow \\infty$, for some $0 < y < \\infty$, then $1/n|V_{p_nn}V^T_{p_nn}| \\rightarrow (1 + y^{\\frac{1}{2}})^2\\sigma^2$ almost surely, where $|A|$ denotes the operator ("induced") norm of $A$. Since $1/n|V_{p_nn}V^T_{p_nn}|$ is the maximum eigenvalue of $1/nV_{p_nn}V^T_{p_nn}$, the result relates to studies on the spectrum of symmetric random matrices.
Keywords
MathematicsCombinatoricsRandom matrixRandom variableEigenvalues and eigenvectorsZero (linguistics)Norm (philosophy)Matrix normOperator (biology)PhysicsStatisticsQuantum mechanics
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Publication Info
- Year
- 1980
- Type
- article
- Volume
- 8
- Issue
- 2
- Citations
- 441
- Access
- Closed
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Cite This
Stuart Geman
(1980).
A Limit Theorem for the Norm of Random Matrices.
The Annals of Probability
, 8
(2)
.
https://doi.org/10.1214/aop/1176994775
Identifiers
- DOI
- 10.1214/aop/1176994775