A Riemannian inexact Newton-CG method for constructing a nonnegative matrix with prescribed realizable spectrum

Abstract

This paper is concerned with the inverse eigenvalue problem of finding a nonnegative matrix such that it has the prescribed realizable spectrum. We reformulate the inverse eigenvalue problem as an under-determined constrained nonlinear matrix equation over several matrix manifolds. Then we propose a Riemannian inexact Newton-CG method for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed method is established under some assumptions. We also extend the proposed method to the case of prescribed entries. Finally, numerical experiments are reported to illustrate the efficiency of the proposed method.

Appendix

In this appendix, we establish some basic properties of the product manifold $\mathbb {R}^{n\times n}\times \mathcal {O}(n)\times \mathcal {V}$ and the differential of G defined in (2.1). We first show that the nonlinear matrix Eq. (2.1) is under-determined for all $n\ge 2$. The dimension of $\mathbb {R}^{n\times n}\times \mathcal {O}(n)\times \mathcal {V}$ is given by

$$\begin{aligned} \dim (\mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V}) = n^2 + \frac{n(n-1)}{2} + |\mathcal {J}|, \end{aligned}$$

where $\mathcal {J}$ is the complementary index set of $\mathcal {I}$ with respect to the index set $\mathcal {N}$, and $|\mathcal {J}|$ is the cardinality of $\mathcal {J}$. Thus

$$\begin{aligned} \dim (\mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V}) > \dim \mathbb {R}^{n\times n}\quad \text{ for } n\ge 2. \end{aligned}$$

Hence, (2.1) is under-determined for all $n\ge 2$.

The tangent space of $\mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V}$ at a point $(S,Q,V)\in \mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V}$ is given by

$$\begin{aligned} \begin{array}{c} T_{(S,Q,V)}\big ( \mathbb {R}^{n\times n}\times \mathcal {O}(n)\times \mathcal {V}\big ) = T_S \mathbb {R}^{n\times n}\times T_Q \mathcal {O}(n)\times T_V \mathcal {V}. \end{array} \end{aligned}$$

Here, $T_S \mathbb {R}^{n\times n}$, $T_Q \mathcal {O}(n)$, and $T_V \mathcal {V}$ are the tangent spaces of $\mathbb {R}^{n\times n}$, $\mathcal {O}(n)$, and $\mathcal {V}$ at $S\in \mathbb {R}^{n\times n}$, $Q\in \mathcal {O}(n)$, and $V\in \mathcal {V}$ accordingly, which are given by [1, p. 42]:

$$\begin{aligned} T_S \mathbb {R}^{n\times n}=\mathbb {R}^{n\times n},\quad T_Q \mathcal {O}(n)= \big \{ Q\varOmega \ | \ \varOmega ^T = -\varOmega ,\; \varOmega \in \mathbb {R}^{n\times n}\big \}, \quad T_V \mathcal {V}= \mathcal {V}. \end{aligned}$$

A retraction R on $\mathbb {R}^{n\times n}\times \mathcal {O}(n)\times \mathcal {V}$ is given by

$$\begin{aligned} R_{(S,Q,V)} (\xi _S,\zeta _Q,\eta _V) = \big (R_S(\xi _S), R_Q(\zeta _Q),R_{V}(\eta _V)\big ) \end{aligned}$$

for all $(S,Q,V) \in \mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V}$ and $(\xi _S,\eta _Q,\gamma _V)\in T_{(S,Q,V)}\big ( \mathbb {R}^{n\times n}\times \mathcal {O}(n)\times \mathcal {V}\big )$, where $R_S$, $R_Q$, and $R_{V}$ are the retractions on $\mathbb {R}^{n\times n}$, $\mathcal {O}(n)$, and $\mathcal {V}$ accordingly, which may take the following form:

$$\begin{aligned} \left\{ \begin{array}{ccl} R_S(\xi _S) &{}=&{} S + \xi _S, \quad \mathrm{for} \; \xi _S \in T_S\mathbb {R}^{n\times n},\\ R_Q(\zeta _Q) &{}=&{} \mathrm{qf} (Q + \zeta _Q), \quad \mathrm{for} \; \zeta _Q \in T_Q\mathcal {O}(n),\\ R_{V}(\eta _V) &{}=&{} V+\eta _V, \quad \mathrm{for} \; \eta _V \in T_{V} \mathcal {V}. \end{array} \right. \end{aligned}$$

Here, $\mathrm{qf}(A)$ means the Q factor of the QR decomposition of a nonsingular matrix $A\in \mathbb {R}^{n\times n}$ in the form of $A=Q\widetilde{R}$ with $Q\in \mathcal {O}(n)$ and $\widetilde{R}$ being an upper triangular matrix with strictly positive diagonal entries. For other choices of retractions on $\mathcal {O}(n)$, one may refer to [1, pp. 58–59].

We now establish the differential of G. By simple calculation, the differential $\mathrm {D}G(S,Q,V): T_{(S,Q,V)}\big (\mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V}\big ) \rightarrow T_{G(S,Q,V)}\mathbb {R}^{n\times n}\simeq \mathbb {R}^{n\times n}$ of G at $(S,Q,V) \in \mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V}$ is determined by

$$\begin{aligned} \mathrm {D}G(S,Q,V) [(\varDelta S,\varDelta Q, \varDelta V)] = 2S\odot \varDelta S + [ Q(\varLambda +V)Q^T , \varDelta QQ^T]- Q\varDelta VQ^T \end{aligned}$$

for all $(\varDelta S,\varDelta Q, \varDelta V) \in T_{(S,Q,V)}(\mathbb {R}^{n\times n}\times \mathcal {O}(n) \times \mathcal {V})$. On the other hand, with respect to the Riemannian metric defined in (2.2), the adjoint $(\mathrm {D}G(S,Q,V))^* : T_{G(S,Q,V)}\mathbb {R}^{n\times n}\rightarrow T_{(S,Q,V)}(\mathbb {R}^{n\times n}\times \mathcal {O}(n)\times \mathcal {V})$ of $\mathrm {D}G(S,Q,V)$ is determined by

$$\begin{aligned}&(\mathrm {D}G(S,Q,V))^* [ \varDelta Z ]\\&\quad = ((\mathrm {D}G(S,Q,V))_1^* [ \varDelta Z ],(\mathrm {D}G(S,Q,V))_2^* [ \varDelta Z ],(\mathrm {D}G(S,Q,V))_3^* [ \varDelta Z ]) \end{aligned}$$

for all $\varDelta Z\in T_{G(S,Q,V)}\mathbb {R}^{n\times n}$, where for each $\varDelta Z\in T_{G(S,Q,V)}\mathbb {R}^{n\times n}$,

$$\begin{aligned} \left\{ \begin{array}{rcl} (\mathrm {D}G(S,Q,V))_1^* [ \varDelta Z ] &{}=&{} 2S\odot \varDelta Z, \\ (\mathrm {D}G(S,Q,V))_2^* [ \varDelta Z ] &{}=&{} \displaystyle \frac{1}{2}\big ( [ Q(\varLambda + V)Q^T, (\varDelta Z)^T ] + [ Q(\varLambda + V)^TQ^T, \varDelta Z ]\big )Q, \\ (\mathrm {D}G(S,Q,V))_3^* [ \varDelta Z ] &{}=&{} -W\odot \big (Q^T\varDelta Z Q\big ). \end{array} \right. \end{aligned}$$

Here, $W\in \mathbb {R}^{n\times n}$ is defined by

$$\begin{aligned} W_{ij}=\left\{ \begin{array}{ll} 0,&{} \text{ if } (i,j) \in \mathcal {I},\\ 1, &{} \text{ otherwise }. \end{array} \right. \end{aligned}$$

(A.1)

Analogously, we can establish the tangent space of the product manifold $\mathcal {Z}\times \mathcal {O}(n)\times \mathcal {V}$, a retraction on $\mathcal {Z}\times \mathcal {O}(n)\times \mathcal {V}$, and the differential of H defined in (4.2) and its adjoint. Here, $\mathcal {Z}\times \mathcal {O}(n)\times \mathcal {V}$ is equipped with the Riemannian metric defined as in (2.2).