= Tensor-product operator
For <linear operators> $A$ on $V$ and $B$ on $W$, the <tensor product> operator is characterized by
$$
(A\otimes B)(v\otimes w)=Av\otimes Bw.
$$
For <bounded linear operators> on <Hilbert spaces>, it extends continuously to the <Hilbert tensor product>. If both factors are <positive semidefinite operators>, so is their <tensor product> operator. In finite dimensions an <orthonormal eigenbasis> in each factor gives a <tensor-product basis> with nonnegative <eigenvalues> $a_ib_j$.
The positivity assertion also holds without an <eigenbasis>. Take the <positive square root of an operator> of each factor and put $C=A^{1/2}\otimes B^{1/2}$. Then $A\otimes B=C^\dagger C$, so its <quadratic form> is $\|Cz\|^2\geq0$. This is the positivity used in the necessary direction of the <positive partial transpose criterion>, where each product term in a separable <density matrix> remains positive after transposing one factor.
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