The necessary condition for eigenvalue values of a circulant matrix is studied It is then proved that the necessary condition also su ces the existence of a circulant matrix with the prescribed eigenvalue values Introduction An n n matrix C of the form C c c cn cn c c cn cn cn c cn c c cn c is called a circulant matrix As each row of a circulant matrix is just the previous row cycled forward on...

We present an algorithm for computing a d-dimensional subspace of the row space of a matrix. For an n×n matrix A with m nonzero entries and with rank(A) ≥ d the algorithm generates a d × n matrix with full row rank and which is a subspace of Rows(A). If rank(A) < d the algorithm generates a rank(A)×n row-equivalent matrix. The running time of the algorithm is

The Hadamard maximal determinant (maxdet) problem is to find the maximum determinant H(n) of a square {+1,−1}-matrix of given order n. Such a matrix with maximum determinant is called a Doptimal design of order n. We consider some cases where n 6= 0 mod 4, so the Hadamard bound is not attainable, but bounds due to Barba or Ehlich and Wojtas may be attainable. If R is a matrix with maximal (or c...

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F , we say that a (0,1)-matrix A has F as a Berge hypergraph if there is a submatrix B of A and some row and column permutation of F , say G, with G 6 B. Letting ‖A‖ denote the number of columns in A, we define the extremal function Bh(m,F ) = max{‖A‖ : A m-rowed simple matrix and no Berge hypergraph F}. We determine...

Associated with any matrix, there are four fundamental subspaces: the column space, row space, (right) null space, and left null space. We describe a single computation that makes readily apparent bases for all four of these subspaces. Proofs of these results rely only on matrix algebra, not properties of dimension. A corollary is the equality of column rank and row rank. Bringing a matrix to r...

In this paper, we present a new sparse matrix data format that leads to improved memory coalescing and more efficient sparse matrix-vector multiplication (SpMV) for a wide range of problems on high throughput architectures such as a graphics processing unit (GPU). The sparse matrix structure is constructed by sorting the rows based on the row length (defined as the number of non-zero elements i...

where subscripts denote partial derivatives, and should be thought of as row vectors, vs. column vectors x and p. So, for example, g x is a 1×K matrix, and f p is aK×P matrix. Equation (1) is derived simply by applying the chain rule to g = g(x,p) = g(f(xn−1,p),p) = g(f(f(xn−2,p),p),p) = · · ·. 1Note that if xn depends on xn−` for ` = 1, . . . , L, the recurrence can still be cast in terms of x...

Suppose that we have a linear program max{cx : Ax ≤ b, x ≥ 0} (P) where A is an m × n matrix, c an n-dimensional row vector, b an m-dimensional column vector, and x an n-dimensional column vector of variables. Now, we add in the restriction that some of the variables in (P) must take integer values. If some but not all variables are integer, we have a Mixed Integer Program (MIP), written as max...