Search

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of such a space when the maximal dimension is attained. Extensions of this result in the direction of linear spaces of matrices with a bounded number of eigenvalues have been studied. In this paper, we answer what is perhaps the most general problem of the kind as proposed by Loewy and Radwan, by solving their conjecture in the positive. We give the maximal dimension of a vector space of $ n\times n $ n × n matrices with no more than k

For every $ n \in \mathbb {N} $ n ∈ N and every field K, let $ A(n,K) $ A (n , K) be the vector space of the antisymmetric $ (n \times n) $ (n × n) -matrices over K. We say that an affine subspace S of $ A(n,K) $ A (n , K) has constant rank r if every matrix of S has rank r. Define $$\begin{align*} {\mathcal{A}}_{antisym}^K(n;r)& = \{ S \;| \; S \; {\rm affine \; subspace \; of } \; A(n,K) \; {\rm of \; constant \; rank } \; r \}, \\ a_{antisym}^K(n;r) & = \max \{\dim S \mid S \in {\mathcal{A}}_{antisym}^K(n;r) \}. \end{align*}$$ A antisym K (n ; r) = { S | S affine subspace of A (n , K) of constant rank r } , a antisym K (n ; r) = max { dim ⁡ S ∣ S ∈ A antisym K (n ; r) }. In this paper, we prove the following formulas: for $ n \geq ~2r +2 $ n ≥ 2 r + 2 \[ a_{antisym}^{\mathbb{R}}(n; 2r) = (n-r-1) r ; \] a antisym R (n ; 2 r) = (n − r − 1) r ; for n = 2r \[ a_{antisym}^{\mathbb{R}}(n; 2r) =r(r-1) ; \] a antisym R (n ; 2 r) = r (r − 1) ; for n = 2r + 1 \[ a_{antisym}^{\mathbb{R}}(n; 2r) = r(r+1). \] a antisym R (n ; 2 r) = r (r + 1). [ABSTRACT FROM AUTHOR]

We propose an effective matrix iteration method and a novel zeroing neural network (NZNN) model for finding numerical commuting solutions of the (time-invariant) Yang–Baxter-like matrix equation in this paper. The proposed matrix iteration method has a second-order convergence speed, and it is proved to be stable. How to proceed with initial value selection and termination criteria are discussed. Meanwhile, two numerical experiments are adopted to illustrate the superiority of the proposed matrix iteration method in computational efficiency. The NZNN model based on Tikhonov regularization for solving the time-invariant Yang–Baxter-like matrix equation is given. Besides, numerical results are provided to substantiate the efficiency, availability and superiority of the developed NZNN model for time-invariant Yang–Baxter-like matrix equation problems. [ABSTRACT FROM AUTHOR]

The block trace and partial trace are closely related. However, the block determinant and partial determinant are quite different. We introduce a new determinant-like operation, the determinant-root, which has a partial operation that is more natural in the algebraic sense. The block determinant retains some of the algebraic properties of determinants, while the partial determinant-root retains the same algebraic properties for matrices which are Kronecker products. We investigate the equations $ \det (D(A)) = \det (A) $ det (D (A)) = det (A) , $ D(AB) = D(A)D(B) $ D (AB) = D (A) D (B) and $ D(e^A) = e^{T(A)} $ D (e A) = e T (A) , where D is the block/partial/partial-root determinant and T is the corresponding block/partial/partial-normalized trace. These results yield a characterization of non-linear preservers of determinants of Kronecker products. [ABSTRACT FROM AUTHOR]