Search results
Results From The WOW.Com Content Network
Then the rank function of M is: r(B) = min(k, |B|). Let M be a partition matroid: the elements of E are partitioned into categories, each category c has capacity k c, and the independent sets are those containing at most k c elements of category c. Then the rank function of M is: r(B) = sum c min(k c, |B c |) where B c is the subset B contained ...
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}} denote the space of n × n {\displaystyle n\times n} matrices.
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the ...
The Symmetric Rank 1 (SR1) method is a quasi-Newton method to update the second derivative (Hessian) based on the derivatives (gradients) calculated at two points. It is a generalization to the secant method for a multidimensional problem. This update maintains the symmetry of the matrix but does not guarantee that the update be positive definite.
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} .
The remaining task has been reduced to finding the eigenvalues of a diagonal matrix plus a rank-one correction. Before showing how to do this, let us simplify the notation. We are looking for the eigenvalues of the matrix D + w w T {\displaystyle D+ww^{T}} , where D {\displaystyle D} is diagonal with distinct entries and w {\displaystyle w} is ...
In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank.
Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be represented by a 6×6 matrix ...