Linear Algebra Done Right · Problem 9D.10

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · tensor-products

Problem

Suppose

S \in \mathcal{L}(V)

and

T \in \mathcal{L}(W)

. Prove that

S \otimes T

is an invertible operator on

V \otimes W

if and only if both

S

and

T

are invertible operators. Also, prove that if both

S

and

T

are invertible operators, then

(S \otimes T)^{-1} = S^{-1} \otimes T^{-1}

, where we are using the notation from the comment after Exercise 9.