# What is the significance of the largest eigenvalue of a matrix?

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?

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

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.
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.