|M.Sc Student||Bortnik Olga|
|Subject||Words in Positive Definite Matrices|
|Department||Department of Applied Mathematics||Supervisor||Professor Emeritus Abraham Berman|
In this research we deal with words in positive definite matrices. A word in positive definite? matrices is a finite product of positive definite matrices with real, positive or natural powers. We call a word in positive definite matrices good if it has only positive eigenvalues for all positive definite matrices, substituted in it. The eigenvalues and trace of words in positive definite matrices will be of our primary interest. In our research we are interested in good words in two positive definite matrices and in good words in several positive definite matrices. We also want to find additional conditions on matrices and on structure of the word, which guarantee that all the eigenvalues of it will be positive. In the work we give classification of words with positive eigenvalues and prove a necessary condition for a word with real exponents to be good. We also consider the polynomial , in which and are two positive definite matrices and? , and discuss the conjecture that the polynomial? has all positive coefficients whenever and are positive definite matrices. In this research we also deal with symmetric word equations in two positive definite matrices and of the form , in which is a symmetric word and all exponents of are positive. We prove the theorem that every positive definite matrix is the product of? the form that for all or , in which and are two positive definite matrices and the matrices are not? roots of the matrix .