The zero-term rank of a matrix is the minimum number of lines (row or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve the zero-term rank of the m × n integer matrices. That is, a linear operator T preserves the zero-term rank if and only if it has the form T(A)=P(A ∘ B)Q, where P, Q are permutation matrices and A ∘ B is the Schur product with B whose entries are all nonzero integers.
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In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.
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In this paper, an approach based on matrix polynomials is introduced for solving linear systems of partial differential equations. The main feature of the proposed method is the computation of the Smith canonical form of the assigned matrix polynomial to the linear system of PDEs, which leads to a reduced system. It will be shown that the reduced one is an independent system of PDEs having only one unknown in each equation. A comparison of the results for several test problems reveals that the method is very effective and convenient. The basic idea described in this paper can be employed to solve other linear functional systems.
In this paper, we obtain some applications of first order differential subordination and superordination results involving certain linear operator and other linear operators for certain normalized analytic functions. Some of our results improve and generalize previously known results.
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We study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.
For a rank-1 matrix $A = a ⊗ b^{t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T(A) = U ⊗ A^{t} ⊗ V$ with some monomial matrices U and V.
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