# Numerical range of doubly stochastic matrices

## Doubly stochastic matrices

A doubly stochastic matrix $A \in \mathbb{R}^{n \times n }$ is a matrix for which the entries are non-negative while the row and column sums are all equal to one $\sum_{j=1}^{n} a_{ij} = 1 \text{ for } i=1,\ldots,n$ and $\sum_{i=1}^{n} a_{ij} = 1 \text{ for } j=1,\ldots,n$

### Theorem

Let A be $4\times 4$ doubly stochastic matrix. Then, $W(A)$consists of elliptical arcs and line segments if and only if$$:= (A) - 1 + is an eigenvalue of A (multiple, if$$\mu =1$$). If, in addition A - 1 - 3>0 , (A -1 - 3)^2 - (A^A) + 1 +2 + ^2 > 0$$

then $W(A)$ also has corner points at $\mu$ and 1, and thus four flat portions on the boundary. Otherwise, 1 is the only corner point of $W(A)$ and $\partial W(A)$ consists of two flat portions and one elliptical arc.

### Example 1

Consider the doubly stochastic matrix : $A = \begin{pmatrix} 0&1/3&1/4&5/12\ 1/3&0&1/2&1/6\ 1/4&9/32&1/4&1/6\ 5/12&37/96&0&19/96 \end{pmatrix}$

Using above theorem, we compute $\mu = -1/3.$ By calculating the characteristic polynomial and computin the conditions from Theorem we have that one of condition is negative, Theorem 1 states that $\partial W(A)$ has two flat portions and one elliptical arc. Indeed, A reduces unitarily to $(1) \oplus A_1$ for some $3\times 3$ matrix $A_1$, and in Fig. 1 we give $W(A)$, and the horizontal ellipse it contains, $W(A_1)$.

Fig. 1 Numerical range of $A$ with two flat portions and an elliptical arc on the boundary, and the ellipse it contains. The eigenvalues of A are indicated by the points.

### Example 2

Consider the doubly stochastic matrix : $A = \begin{pmatrix} 0&1/3&1/4&5/12\ 1/3&0&1/2&1/6\ 1/4&1/8&1/4&3/8\ 5/12&13/24&0&1/24 \end{pmatrix}$ By coincidence, we again compute $\mu = -1/3$, and though the characteristic polynomial again has $\mu$ as a root. The formulas in inequalities on Theorem evaluate to $7/24$ and $59/576$ respectively, so the number of flat portions is still the same. However, Fig. 2 shows that $W(A)$ is the convex hull of a vertical ellipse and the point 1, as opposed to the horizontal ellipse in the previous example. Indeed, the eigenvalues marked in the graph include a complex conjugate pair.

Fig. 2 Numerical range $A$ with two flat portions and an elliptical arc on the boundary, and the contained vertical ellipse. The eigenvalues of $A$ are indicated by the points.

This section is created based on [1] in which we can find more examples also.

# References

1. [1]K. A. Camenga, P. X. Rault, D. J. Rossi, T. Sendova, and I. M. Spitkovsky, “Numerical range of some doubly stochastic matrices,” Applied Mathematics and Computation, vol. 221, pp. 40–47, 2013, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0096300313006231.