Advanced Linear Algebra, Lecture 7.1: Definiteness and indefiniteness

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Advanced Linear Algebra, Lecture 7.1: Definiteness and indefiniteness

A matrix M is positive-definite, or just positive, if (x,Mx) is positive for all nonzero x. We can similarly define what it means to be nonnegative, negative, and nonpositive. These are equivalent to the eigenvalues of M being positive, nonnegative, negative, and nonpositive, respectively. A matrix is said to be indefinite if it is none of these, i.e., if it has both positive and negative eigenvalues. We prove some basic properties about positive maps, such as that they always have a unique square root. Finally, we show that in the space of self-adjoint maps, the set of positive maps is open, and its closure are the nonnegative maps.