The matrix equation X + ATX−1A = Q has been studied extensively when A and Q are real square matrices and Q is symmetric positive definite. The equation has positive definite solutions under suitable conditions, and in that case the solution of interest is the maximal positive definite solution. The same matrix equation plays an important role in Green’s function calculations in nano research, ...

for all complex vectors ai and bj . One can easily prove that if X is positive definite then X is hermitian (see, e.g., Ref. [1], p. 65). Since the eigenvalues of hermitian matrices are real, it is easy to prove that the eigenvalues of positive definite matrices are real and positive. Moreover, a positive definite matrix is invertible, since it does not possess a zero eigenvalue. Note that a no...

We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: On-line learning with a simple square loss and finding a symmetric positiv...

The problem we are interested in is the best approximation of a given matrix by a positive semi–definite symmetric Toeplitz matrix. Toeplitz matrices appear naturally in a variety of problems in engineering. Since positive semi–definite Toeplitz matrices can be viewed as shift invariant autocorrelation matrices, considerable attention has been paid to them, especially in the areas of stochastic...