Symmetric nonnegative realization of spectra
... nonnegative matrices. For n ≥ 5 the problem remains unsolved. In the general case, when the possible spectrum Λ is a set of complex numbers, the problem has only been solved for n = 3 by Loewy and London [11]. The complex cases n = 4 and n = 5 have been solved for matrices of trace zero by Reams [17 ...
... nonnegative matrices. For n ≥ 5 the problem remains unsolved. In the general case, when the possible spectrum Λ is a set of complex numbers, the problem has only been solved for n = 3 by Loewy and London [11]. The complex cases n = 4 and n = 5 have been solved for matrices of trace zero by Reams [17 ...
4 Images, Kernels, and Subspaces
... (a) Recall that ker(AB) is the set of vectors x ∈ Rm for which ABx = 0, and similarly that ker(B) is the set of vectors x ∈ Rm for which Bx = 0. Now if x is in ker(B), then Bx = 0, so ABx = 0. This means that x is in ker(AB), so we see that ker(B) must always be contained in ker(AB). On the other h ...
... (a) Recall that ker(AB) is the set of vectors x ∈ Rm for which ABx = 0, and similarly that ker(B) is the set of vectors x ∈ Rm for which Bx = 0. Now if x is in ker(B), then Bx = 0, so ABx = 0. This means that x is in ker(AB), so we see that ker(B) must always be contained in ker(AB). On the other h ...
on the complexity of computing determinants
... our description we also take into account the interpretation by Villard (1997a; 1997b), where the relevant literature from linear control theory is cited. Our algorithms rely on the notion of minimum linear generating polynomials (generators) of matrix sequences. This notion is introduced below in S ...
... our description we also take into account the interpretation by Villard (1997a; 1997b), where the relevant literature from linear control theory is cited. Our algorithms rely on the notion of minimum linear generating polynomials (generators) of matrix sequences. This notion is introduced below in S ...
Algorithms and Applications for Approximate Nonnegative Matrix
... “approximate” in the remainder of this paper. An appropriate decision on the value of k is critical in practice, but the choice of k is very often problem dependent. In most cases, however, k is usually chosen such that k ≪ min(m, n) in which case WH can be thought of as a compressed form of the dat ...
... “approximate” in the remainder of this paper. An appropriate decision on the value of k is critical in practice, but the choice of k is very often problem dependent. In most cases, however, k is usually chosen such that k ≪ min(m, n) in which case WH can be thought of as a compressed form of the dat ...
Efficient Dense Gaussian Elimination over the Finite Field with Two
... AMD Opteron. Thus, we assume in this work that native CPU words have 64 bits. However, it should be noted that our code also runs on 32-bit CPUs and on non-x86 CPUs such as the PowerPC. Element-wise operations over F2 , being mathematically trivial, are relatively cheap compared to memory access. In ...
... AMD Opteron. Thus, we assume in this work that native CPU words have 64 bits. However, it should be noted that our code also runs on 32-bit CPUs and on non-x86 CPUs such as the PowerPC. Element-wise operations over F2 , being mathematically trivial, are relatively cheap compared to memory access. In ...
