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The Tracy-Widom distribution gives the limiting distribution of the largest eigenvalue of a random matrix (in the $\beta$-Hermite ensemble, where $\beta$ is 1,2 or 4).

The second smallest eigenvalue of the Laplacian helps you divide the graph into communities, known as the algebraic connectivity...

But what is so important about the largest eigenvalue of a matrix? Is it related to geometry? Or dynamics, where such a matrix may have a direct meaning?

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    $\begingroup$ What if the eigenvalues are complex? That does not happen in the case of the Laplacian, since it's symmetric. $\endgroup$ May 6, 2019 at 17:01

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The following excerpt is taken from the book Spectra and pseudospectra of Trefethen and Embree, first page.

In applications, eigenvalues are typically used where a matrix is to be compounded iteratively, for example as a power $A^k$ or an exponential $$ e^{tA}=I+tA+\frac12 t^2A^2 + \frac1{3!}t^3A^3+\ldots$$

If the eigenvalues of $A$ are $\lambda_1, \lambda_2,\ldots,\lambda_n$, then the eigenvalues of $A^k$ are, unsurprisingly, $\lambda_1^k, \lambda_2^k, \ldots, \lambda_n^k$. In particular, they have modulus $$ |\lambda_1|^k, |\lambda_2|^k, \ldots, |\lambda_n|^k.$$ Now suppose that $|\lambda_1|>|\lambda_j|$ for all $j=2,\ldots,n$. Then, obviously, $$ \frac{|\lambda_j|^k}{|\lambda_1|^k}\to 0.$$ This shows that, as $k\to \infty$, the non-dominant eigenvalues are asymptotically negligible. Thus, they have lesser importance in any situation in which the computation of $A^k$ is required. As Trefethen and Embree point out, these situations are the most important in applications.

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The largest eigenvalue (in absolute value) of a normal matrix is equal to its operator norm.

So, for instance, if $A$ is a square matrix with largest eigenvalue $\lambda_{\mathrm{max}}$, and $x$ is a vector, you know that $\|Ax\| \le |\lambda_{\mathrm{max}}| \|x\|$, and this is sharp (here $\|\cdot\|$ is the usual Euclidean norm). This is very useful in many kinds of estimates.

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    $\begingroup$ @RodrigodeAzevedo: I meant to say "normal matrix". In that case, the singular values are the absolute values of the eigenvalues. This question is probably most natural when dealing with positive definite matrices; then the singular values and eigenvalues are the same. $\endgroup$ May 6, 2019 at 17:06

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