Class of finite-dimensional matrices with diagonals that majorize their spectrum

1 Introduction

Majorization provides a preorder relational operator for comparing the relative disorder (entropy) of the distribution of elements of two vectors. Specifically, given two vectors x , y ∈ R n , whose respective elements sum to unity^[1], then y is said to majorize x , notated as x ≺ y , if and only if there exists a set of permutation matrices P j and probabilities p j , such that

This expression of x as a probabilistic sum of permutations of y can be interpreted as indicating that the distribution of elements of x is more disordered than that of y . An alternative definition is that x ≺ y if and only if

where S is a doubly stochastic matrix, i.e., nonnegative with every row and column sum equal to unity. This definition makes the probabilistic interpretation even more explicit in the form S y → x , where x is seen to be a conservative stochastic evolutionary state of y . This can be made rigorous by the following known result:

where H ( ⋅ ) is the Shannon entropy, H ( z ) ≐ − ∑ i = 1 n z i log z i , where z i log z i is taken as 0 for z i = 0 .

The Schur-Horn theorem [2,8] says that every Hermitian matrix has the property that its spectrum (vector of eigenvalues) majorizes its diagonal, and the same can be inferred more generally from equation (2) for all normal, i.e., unitarily diagonalizable, matrices. This structural property immediately provides an additional means for analyzing, modeling, and controlling systems involving such matrices. A prominent example from physics is the density matrix describing the quantum state of a system, which is Hermitian (more specifically, positive semidefinite) and therefore normal.

In the case of a normal matrix, the majorization property implies that its diagonal provides an entropic upper bound on the distribution of its eigenvalues. Having such a link between the diagonal and spectrum of a given matrix offers a variety of potential theoretical and practical benefits, but heretofore there has been no other matrix class for which such a majorization property has been proven to hold.

In Section 2, we identify a new finite-dimensional matrix class for which an alternative diagonal/spectrum majorization relationship can be proven to exist.

2 Special PD-diagonalizable matrices

We begin by stating a previously studied [13] conjecture/result:

PD-IRGA conjecture: Given any real n × n symmetric positive-definite (PD) matrix P , then for n ≤ 6 :

where “ ∘ ” is the Schur-Hadamard elementwise matrix product.

Such a result would be remarkable because it implies that important properties exist for low-dimensional PD matrices that do not hold generally. Specifically, it can be shown by counterexample that nonnegativity does not generally hold for n > 6 , and it has been proven for all n ≤ 4 (see ref. [13] for details).

Finite-dimensional properties of this kind are of particular interest when analyzing systems that are intrinsically low dimensional due to constraints imposed by physical theory, e.g., the three spatial dimensions of the physical world. Here, however, we focus on the appearance of the doubly stochastic matrix S and observe that it can permit us to define a special finite-dimensional matrix class for which the diagonal majorizes the spectrum. Specifically, letting k be the largest integer for which the PD-IRGA conjecture holds, we define the following matrix form:

The motivation for this definition is the following:

The proof relies on the known result [9] that for a diagonalizable matrix B = A E A − 1 :

where A − T is the transposed^[2] inverse of A . In other words, the matrix A ∘ A − T – which may be singular even if A is not – maps the spectrum of B to the diagonal of B . If A is real symmetric PD, then A ∘ A − 1 is nonsingular, so multiplying both sides by its inverse S = ( A ∘ A − 1 ) − 1 gives

where S is doubly stochastic. Thus, by the majorization criterion of equation (2), it can be inferred that the diagonal of a special PD-diagonalizable matrix B majorizes its spectrum:

and, therefore, its entropy gives a lower bound on that of its spectrum, i.e.,

3 Potential applications

General PD-similarity transformations appear in a variety of theoretical and practical applications. These include Gaussian orthogonal ensembles from random matrix theory [10] and their applications to pseudo/quasi-Hermitian operators in physics [5,12], e.g., Su-Schrieffer-Heeger models with real eigenvalues, for which a non-Hermitian Hamiltonian H is mapped to a Hermitian Hamiltonian H ˆ by a PD similarity transformation [3,4,11]. It can be observed that for real^[3] H and H ˆ , a real orthogonal transformation of the system that diagonalizes H ˆ will transform H to PD-diagonalizable form. This can be shown by expressing the normal matrix H ˆ as follows:

and its PD-similar companion H as follows:

Applying the diagonalizing similarity transformation U to H gives

where P = ( U T V ) D ( U T V ) T . If n ≤ k , then U T H U is special PD diagonalizable and, therefore, its diagonal majorizes its spectrum. In other words, under these conditions, the matrix H can be orthogonally transformed so that its diagonal majorizes the spectrum of the potentially unknown Hamiltonian H ˆ . Alternatively, the process can be viewed from an optimization perspective as a problem of identifying an orthogonal-similarity transformation W such that the Shannon entropy of diag ( WHW T ) is optimized via majorization to provide a best-possible provable bound on the entropy of its spectrum.

The PD-IRGA conjecture originated in the area of control theory [13], and special PD-diagonalizable matrices can be examined in this context with the spectrum representing the state of a system. It should be noted that the intrinsic representable dimensionality of this state is not necessarily limited to k ≤ 6 when eigenvalues provide additional degrees of freedom. Specifically, while the doubly stochastic mapping from spectrum to diagonal in equation (6) requires the diagonalizing matrix to be real PD, that condition is not required for the spectrum if majorization is generalized (see ref. [2], p. 110) from vectors over R n to C n , and more generally to their normed division-algebra extensions over H n and O n . Thus, if the spectrum is defined over the latter, i.e., octonions, then the maximum representable dimensionality is 8 ⋅ k , which gives 48 degrees of freedom for the conjectured value of k = 6 .

Generalizations of majorization to state vectors over hypercomplex spaces (or other algebras, e.g., ∗ -algebras) have important implications for practical applications when computation of the spectrum is prohibitively expensive. For example, consider a measurement process that is designed to obtain a special PD-diagonalizable matrix X and normal matrix Y , both having the same spectrum, i.e., the state distribution of interest. If it is not practical to compute the spectrum explicitly, then Theorem 1 implies that lower and upper entropic bounds on the measured state can be obtained from the diagonals of X and Y , respectively. These bounds may be sufficient for the needs of the application or, if not, may provide a means for guiding a search of the space toward the case in which the measurement matrices become equal, i.e., diagonal and give the exact spectrum of the target state.

4 Discussion

The principal contribution of this article is the definition of a new class of finite-dimensional matrices, special PD-diagonalizable, for which there exists a majorization relationship between the diagonal and the spectrum. The definition of this class exploits the proven cases of the PD-IRGA conjecture, which presently is restricted to a subset of n × n matrices with n ≤ 4 , but is conjectured to hold for n ≤ 6 .

Future work will examine applications of these results to tracking and control problems in which measurements of an evolving state are transformed by PD similarity with respect to the covariant derivative along a Riemannian manifold [6,7], which may also be applicable to models arising in both theoretical and applied physics. Of particular focus for the latter will be potential applications to robust quantum circuits [14]. From a theoretical physics perspective, understanding the structural constraints imposed by dimensionality d ≤ 4 or d ≤ 6 that guarantee the properties of special PD-diagonalizable matrices may reveal deeper insights relating to quantum field theories that are strictly constrained to dimensions d = 4 or d = 6 . (e.g., conformal field theory [15] and loop quantum gravity [1]).