Abstract: Let $$M_{n} (\mathbb{C})$$ be the algebra of all $$n$$ x $$n$$ complex matrices. For any $$X∈Mn(\mathbb{C})$$, the real part of $$ X$$ is $$Re X=\frac{X+X^{*}}{2} $$and the imaginary part of $$X$$ is $$Im X=\frac{X-X^{*}}{2i}$$. Let $$Υ,Ƥ$$, and $$Ψ∈Mn(\mathbb{C})$$ be accretive-dissipative matrices such that the real parts and imaginary parts of their Cartesian decompositions are positive semidefinite. In this paper, we present and prove the following determinant inequalities. <br> $$2^{\frac{n}{2}}|det(Υ+Ƥ+Ψ)|+2^{\frac{n}{2}}|det(Ψ)|−(|det(Υ+Ψ)|+|det(Ƥ+Ψ)|)≥2^{-\frac{n}{2}}|det(Υ+Ƥ)|−(|det(Υ)|+|det(Ƥ)|)$$, $$|det(Υ+Ƥ+Ψ)|+|det(Υ)|+|det(Ƥ)|+|det(Ψ)|≥2^{-\frac{n}{2}}(|det(Υ+Ƥ)|+|det(Υ+Ψ)|+|det(Ƥ+Ψ)|)$$,<br> and $$|det(Υ+Ƥ+Ψ)|+|det(Ψ)|≥2^{-\frac{n}{2}}(|det(Υ+Ψ)|+|det(Ƥ+Ψ)|+(2^{n}−2)|det(Υ)|^{\frac{1} {2}}$$ $$|det(Ƥ)|^{\frac{1}{2}}+3(3^{n−1}−2^{n}+1)\sqrt[3]{|det(Υ)||det(Ƥ)||det(Ψ)|3})$$. Moreover, numerical example is tested to these inequalities.

Keywords: Positive semidefinite matrix, Accretive-dissipative matrices, Determinant inequalities, norm, Cartesian decomposition, complex matrices

Pages: 450-453