84 2 Algebra
Solution.By the Cayley–Hamilton Theorem,
B^2 −(trB)B+I 2 =O 2.Multiply on the left byAB−^1 to obtain
AB−(trB)A+AB−^1 =O 2 ,and then take the trace to obtain the identity from the statement.
Five more examples are left to the reader.258.LetAbea2×2 matrix. Show that if for some complex numbersuandvthe matrix
uI 2 +vAis invertible, then its inverse is of the formu′I 2 +v′Afor some complex
numbersu′andv′.
259.Find the 2×2 matrices with real entries that satisfy the equation
X^3 − 3 X^2 =
(
− 2 − 2
− 2 − 2
)
.
260.LetA, B, C, Dbe 2×2 matrices. Prove that the matrix[A, B]·[C, D]+[C, D]·
[A, B]is a multiple of the identity matrix (here[A, B]=AB−BA, the commutator
ofAandB).
261.LetAandBbe 3×3 matrices. Prove that
det(AB−BA)=tr((AB−BA)^3 )
3.
262.Show that there do not exist real 2×2 matricesAandBsuch that their commutator
is nonzero and commutes with bothAandB.
Here is the simplest version of the other result that we had in mind.The Perron–Frobenius Theorem.Any square matrix with positive entries has a unique
eigenvector with positive entries(up to a multiplication by a positive scalar),and the
corresponding eigenvalue has multiplicity one and is strictly greater than the absolute
value of any other eigenvalue.
Proof.The proof uses real analysis. LetA=(aij)ni,j= 1 ,n≥1. We want to show that
there is a uniquev∈[ 0 ,∞)n,v =0, such thatAv=λvfor someλ. Of course, sinceA
has positive entries andvhas positive coordinates,λhas to be a positive number. Denote
byKthe intersection of[ 0 ,∞)nwith then-dimensional unit sphere. Reformulating the
problem, we want to show that the functionf:K→K,f(v)=‖AvAv‖has a fixed point.