Fun with idempotent matrices

Let AA and BB be n×nn\times n commuting, idempotent matrices such that A−BA - B is invertible. Prove that A+BA + B is the n×nn\times n identity matrix.

(Mathematics Magazine Problem Q987 (2009).)

#Matrices #LinearAlgebra #ProblemSolving

Note by Scott Kominers
7 years, 1 month ago

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Comments

Start (A+B)(A−B)=A2−B2−AB+BA=A2−B2+[B,A]\begin{aligned}\left(\mathbf{A}+\mathbf{B}\right)\left(\mathbf{A}-\mathbf{B}\right) &= \mathbf{A}^2-\mathbf{B}^2-\mathbf{A}\mathbf{B}+\mathbf{B}\mathbf{A}\\ &=\mathbf{A}^2-\mathbf{B}^2+\left[\mathbf{B},\mathbf{A}\right]\end{aligned}

That A\mathbf{A} commutes with B\mathbf{B} is to say that [B,A]=BA−AB=0\left[\mathbf{B},\mathbf{A}\right]=\mathbf{B}\mathbf{A} - \mathbf{A}\mathbf{B}=0.

Idempotency is to say A2=A\mathbf{A}^2=\mathbf{A} and likewise for B\mathbf{B}, so

(A+B)(A−B)=A−B\left(\mathbf{A}+\mathbf{B}\right)\left(\mathbf{A}-\mathbf{B}\right) =\mathbf{A}-\mathbf{B}

Acting with the right inverse of A−B\mathbf{A}-\mathbf{B}, which is stated to exist, we have

(A+B)(A−B)(A−B)−1=(A−B)(A−B)−1(A+B)=1\begin{aligned}\left(\mathbf{A}+\mathbf{B}\right)\left(\mathbf{A}-\mathbf{B}\right)\left(\mathbf{A}-\mathbf{B}\right)^{-1} &= \left(\mathbf{A}-\mathbf{B}\right)\left(\mathbf{A}-\mathbf{B}\right)^{-1} \\ \left(\mathbf{A}+\mathbf{B}\right) &= \mathbb{1}\end{aligned}

Josh Silverman Staff - 7 years, 1 month ago
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