Complexity of matrix problems

Abstract

In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying three-valent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated.Comment: 24 page

Classification of the entangled statesL×N×N

“…There the eigenvalues of the Jordan forms serve as the continuous parameters in the entanglement classes [3]. Despite the great progress made in 2 × M × N systems, in many classification schemes the classification of 3 × N × N turns out to be a "wild" problem [5]. Attempts toward the classification of a multipartite system in literature mainly concentrate on its tensor ranks or local ranks of the tensor form of the quantum state, and there is still no systematic method for constructing the canonical form of L × N × N under SLOCC in literature to our knowledge.…”

Normal form of m-by-n-by-2 matrices for equivalence

“…where R, S, and T are the matrices (2 We denote the m-by-n zero matrix by 0 mn . The numbers m and n may be zero: the matrices 0 m0 and 0 0n represent the linear mappings 0 → F m and F n → 0.…”

Canonical forms for unitary congruence and *congruence

“…. , n q , n q are their respective multiplicities; the parameters µ j in (2) are determined by A up to replacement by µ J sj (γ j ) ⊕ J sj (γ −1 j ) , β k , γ j ∈ C, |β k | = 1, 0 < |γ j | < 1.…”

“…For our analysis it is convenient to separate the eigenvalue pairs {−1, −1} of A from the reciprocal pairs of its other eigenvalues in (2). Any unitary similarity that puts A in the diagonal form (2) induces a unitary congruence of A that puts it into a special block diagonal form.…”

“…A problem involving complex matrices is said to be unitarily wild if it contains the problem of classifying arbitrary square complex matrices under unitary *congruence. Since the latter problem contains the problem of classifying an arbitrary system of linear mappings on unitary spaces [22,Section 2.3], it is reasonable to regard any unitarily wild problem as hopeless (by analogy with nonunitary matrix problems that contain the problem of classifying pairs of matrices under similarity [2]). …”