On the determinant of a sum of matrices

@misc{citeulike:1639335, abstract = {Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square matrices and a lot of its properties are very well-known. For instance, the determinant is a multiplicative function, i.e. det(AB)=detA detB, but it is not, in general, an additive function. Another interesting function in the matrix analysis is the characteristic polynomial -- in fact, given a matrix A, this function is defined by \$p\_A(t)=det(tI-A)\$ where I is the identity matrix -- which elements are, up a sign, the elementary symmetric functions associated to the eigenvalues of the matrix A. In the present paper new expressions related with the determinant of sum of matrices and the elementary symmetric functions are given. Moreover, the connection with the Mobius function and the partial ordered sets (poset) is presented. Finally, a problem related with the determinant of sum of matrices is solved.}, author = {Costas-Santos, R. S. }, citeulike-article-id = {1639335}, eprint = {math/0612464}, keywords = {math}, month = {Sep}, posted-at = {2007-09-10 02:54:37}, priority = {2}, title = {On the determinant of a sum of matrices}, url = {http://arxiv.org/abs/math/0612464}, year = {2007} }

TY - ELEC ID - citeulike:1639335 L3 - citeulike-article-id:1639335 TI - On the determinant of a sum of matrices N2 - Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square matrices and a lot of its properties are very well-known. For instance, the determinant is a multiplicative function, i.e. det(AB)=detA detB, but it is not, in general, an additive function. Another interesting function in the matrix analysis is the characteristic polynomial -- in fact, given a matrix A, this function is defined by $p_A(t)=det(tI-A)$ where I is the identity matrix -- which elements are, up a sign, the elementary symmetric functions associated to the eigenvalues of the matrix A. In the present paper new expressions related with the determinant of sum of matrices and the elementary symmetric functions are given. Moreover, the connection with the Mobius function and the partial ordered sets (poset) is presented. Finally, a problem related with the determinant of sum of matrices is solved. KW - math AU - Costas-Santos, RS PY - 2007/Sep/06/ UR - http://arxiv.org/abs/math/0612464 ER -