Abstract
The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem.
Highlights
As an important branch of mathematics, matrix theory has been widely applied in the fields of mathematics and technology, such as optimization theory ([1]), differential equations ([2]), numerical analysis, operations ([3]) and quantum theory ([4])
For any A ∈ Hn, we have A = ∑in=1 λiPi, where λi is the eigenvalue of A and ∑in=1 Pi = Id, PiPj = 0(i = j); specially, when x∗ Ax ≥ 0 for any x ∈ Cn, we denote A ∈ Hn+ (Hn+ is the set of n × n positive-definite Hermitian matrices whose eigenvalues are nonnegative)
Let f be a function with the domain (0, +∞); for any A ∈ Hn+, the matrix function is defined as n f (A) = ∑ fPi. i=1
Summary
As an important branch of mathematics, matrix theory has been widely applied in the fields of mathematics and technology, such as optimization theory ([1]), differential equations ([2]), numerical analysis, operations ([3]) and quantum theory ([4]). In this manuscript, let Cn be an n-dimensional complex vector space with the inner product x, y = x∗y = ∑in=1 xi∗yi for x = (x1, · · · , xn) , y = (y1, · · · , yn) ∈ Cn, where the superscripts x∗ and denote the conjugated transpose of x and the matrix transpose, respectively.
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