in this note we characterize polynomial numerical hulls of matrices $a in m_n$ such that$a^2$ is hermitian. also, we consider normal matrices $a in m_n$ whose $k^{th}$ power are semidefinite. for such matriceswe show that $v^k(a)=sigma(a)$.

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1 = (c+ c )/2 and b2 = (c− c)/2i. Also write B for the spectral scale of {b1, b2} relative to τ . In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information ...

In this study‎, ‎a new polynomial rank transmutation is proposed with the help of‎ ‎ the idea of quadratic rank transmutation mapping (QRTM)‎. ‎This polynomial rank‎ ‎ transmutation is allowed to extend the range of the transmutation parameter from‎ ‎  [-1,1] to [-1,k]‎‎. ‎At this point‎, ‎the generated distributions gain more&lrm...

Abstract. Any given nonnegative matrix A ∈ R can be expressed as the product A = UV for some nonnegative matrices U ∈ R and V ∈ R with k ≤ min{m, n}. The smallest k that makes this factorization possible is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the nonnegative rank is known a priori, no simple ...

Suppose that c is a linear operator acting on an n-dimensional complex Hilbert Space H, and let τ denote the normalized trace on B(H). Set b1 = (c + c )/2 and b2 = (c − c)/2i, and write B for the the spectral scale of {b1, b2} with respect to τ . We show that B contains full information about Wk(c), the k-numerical range of c for each k = 1, . . . , n. We then use our previous work on spectral ...

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given m × n matrix A by a matrix B of rank at most k which is much smaller than m and n. The best rank k approximation can be determined via the singular value decomposition which, however, has prohibitively high computational complexity and storage requirements for very large m and...

 ‎Let $V$ be an $n$-dimensional complex inner product space‎. ‎Suppose‎ ‎$H$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎Hrightarrow mathbb{C} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎Denote by $V_{chi}(H)$ the symmetry class‎ ‎of tensors associated with $H$ and $chi$‎. ‎Let $K(T)in‎ (V_{chi}(H))$ be the operator induced by $Tin‎ ‎text{End}(V)$‎. ‎Th...

We consider the problem of computing low-rank approximations of matrices. The novel aspects of our approach are that we require the low-rank approximations be written in a factorized form with sparse factors and the degree of sparsity of the factors can be traded oo for reduced reconstruction error by certain user determined parameters. We give a detailed error analysis of our proposed algorith...

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given m × n matrix A by a matrix B of rank at most k which is much smaller than m and n. The best rank k approximation can be determined via the singular value decomposition which, however, has prohibitively high computational complexity and storage requirements for very large m and...

Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. This dissertation extends the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix compl...