The nearest ‘doubly stochastic’ matrix to a real matrix with the same first moment
✍ Scribed by William Glunt; Thomas L. Hayden; Robert Reams
- Publisher
- John Wiley and Sons
- Year
- 1998
- Tongue
- English
- Weight
- 52 KB
- Volume
- 5
- Category
- Article
- ISSN
- 1070-5325
No coin nor oath required. For personal study only.
✦ Synopsis
Let T be an arbitrary n × n matrix with real entries. We consider the set of all matrices with a given complex number as an eigenvalue, as well as being given the corresponding left and right eigenvectors. We find the closest matrix A, in Frobenius norm, in this set to the matrix T . The normal cone to a matrix in this set is also obtained. We then investigate the problem of determining the closest 'doubly stochastic' (i.e., Ae = e and e T A = e T , but not necessarily non-negative) matrix A to T , subject to the constraints e T 1 A k e 1 = e T 1 T k e 1 , for k = 1, 2, ... A complete solution is obtained via alternating projections on convex sets for the case k = 1, including when the matrix is non-negative.