62 2 Algebra
The equalityMm+n=MmMnwritten in explicit form is
(
Fm+n+ 1 Fm+n
Fm+n Fm+n− 1)
=
(
Fm+ 1 Fm
Fm Fm− 1)(
Fn+ 1 Fn
Fn Fn− 1)
.
We obtain the identity by setting the upper left corners of both sides equal.
Here are some problems for the reader.199.LetMbe ann×ncomplex matrix. Prove that there exist Hermitian matricesA
andBsuch thatM=A+iB. (A matrixXis called Hermitian ifXt=X).
200.Do there existn×nmatricesAandBsuch thatAB−BA=In?
201.LetAandBbe 2×2 matrices with real entries satisfying(AB−BA)n=I 2 for
some positive integern. Prove thatnis even and(AB−BA)^4 =I 2.
202.LetAandBbe twon×nmatrices that do not commute and for which there exist
nonzero real numbersp, q, rsuch thatpAB+qBA=InandA^2 =rB^2. Prove
thatp=q.
203.Leta, b, c, dbe real numbers such thatc =0 andad−bc=1. Prove that there
existuandvsuch that
(
ab
cd
)
=
(
1 −u
01)(
10
c 1)(
1 −v
01)
.
204.Compute thenth power of them×mmatrix
Jm(λ)=⎛
⎜
⎜
⎜⎜
⎜⎜
⎜
⎝
λ 10 ··· 0
0 λ 1 ··· 0
00 λ··· 0
..
...
.
..
.
... ..
.
000 ··· 1
000 ···λ⎞
⎟
⎟
⎟⎟
⎟⎟
⎟
⎠
,λ∈C.205.LetAandBben×nmatrices with real entries satisfying
tr(AAt+BBt)=tr(AB+AtBt).Prove thatA=Bt.