I-AB is invertible if and only if I-BA is invertible

In this entry A and B are endomorphismsPlanetmathPlanetmath of a vector spaceMathworldPlanetmath V. If V is finite dimensional, we may choose a basis and regard A and B as square matricesMathworldPlanetmath of equal dimensionPlanetmathPlanetmath.

TheoremMathworldPlanetmath - Let A and B be endomorphisms of a vector space V. We have that

  1. 1.

    I-AB is invertiblePlanetmathPlanetmathPlanetmath (http://planetmath.org/LinearIsomorphism) if and only if I-BA is invertible, and moreover

  2. 2.

    I-AB is injectivePlanetmathPlanetmath if and only if I-BA is injective.

Proof :

  1. 1.

    Suppose that I-AB is invertible. We shall prove that B(I-AB)-1A+I is the inversePlanetmathPlanetmathPlanetmathPlanetmath of I-BA. In fact

    (I-BA)(B(I-AB)-1A+I) = B(I-AB)-1A+I-BAB(I-AB)-1A-BA
    = B((I-AB)-1-AB(I-AB)-1)A+I-BA
    = B((I-AB)(I-AB)-1)A+I-BA
    = BA+I-BA
    = I

    A similarPlanetmathPlanetmath computation shows that (B(I-AB)-1A+I)(I-BA)=I, i.e. I-BA is invertible.

    Exchanging the roles of A and B we can prove the ”if” part. So I-AB is invertible if and only if I-BA is invertible.

  2. 2.

    Let us first recall that a linear map between vector spaces is invertible if and only if its kernel ker is the zero vector (see this page (http://planetmath.org/KernelOfALinearTransformation)).

    Suppose I-AB is not injective, i.e. there exists u0 such that (I-AB)u=0. Then


    i.e. Buker(I-BA). Notice that Bu0 because u=ABu (by definition of u), so I-BA is also not injective.

    Similarly, if I-BA is not injective then I-AB is not injective.

Remark - It is known that for finite dimensional vector spaces a linear endomorphism is invertible if and only if it is injective. This does not remain true for infinite dimensional spaces, hence 1 and 2 are two different statements.


The result stated in 1 can be proven in a more general context — If A and B are elements of a ring with unity, then I-AB is invertible if and only if I-BA is invertible. See the entry on techniques in mathematical proofs, in which this result is proven using several different techniques.

This entry is based on http://planetmath.org/?op=getmsg&id=5088this discussion on PM.

Title I-AB is invertible if and only if I-BA is invertible
Canonical name IABIsInvertibleIfAndOnlyIfIBAIsInvertible
Date of creation 2013-03-22 14:44:43
Last modified on 2013-03-22 14:44:43
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 16
Author asteroid (17536)
Entry type Theorem
Classification msc 16B99
Classification msc 15A04
Classification msc 47A10
Related topic TechniquesInMathematicalProofs