MathDB

4

Part of 1994 IMC

Problems(2)

IMC 1994 D1 P4

Source:

3/6/2017
Let α∈R\{0}\alpha\in\mathbb R\backslash \{ 0 \} and suppose that FF and GG are linear maps (operators) from Rn\mathbb R^n into Rn\mathbb R^n satisfying F∘G−G∘F=αFF\circ G - G\circ F=\alpha F.
a) Show that for all k∈Nk\in\mathbb N one has Fk∘G−G∘Fk=αkFkF^k\circ G-G\circ F^k=\alpha kF^k.
b) Show that there exists k≥1k\geq 1 such that Fk=0F^k=0.
IMClinear algebra
IMC 1994 D2 P4

Source:

3/6/2017
Let AA be a n×nn\times n diagonal matrix with characteristic polynomial (x−c1)d1(x−c2)d2…(x−ck)dk(x-c_1)^{d_1}(x-c_2)^{d_2}\ldots (x-c_k)^{d_k} where c1,c2,…,ckc_1, c_2, \ldots, c_k are distinct (which means that c1c_1 appears d1d_1 times on the diagonal, c2c_2 appears d2d_2 times on the diagonal, etc. and d1+d2+…+dk=nd_1+d_2+\ldots + d_k=n).
Let VV be the space of all n×nn\times n matrices BB such that AB=BAAB=BA. Prove that the dimension of VV is d12+d22+⋯+dk2d_1^2+d_2^2+\cdots + d_k^2
IMClinear algebramatrix