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Suppose π΄ and π΅ are upper-triangular matrices of the same size, with
$\alpha_1$ , ..., $\alpha_n$ on the diagonal of π΄ and π½_1 , ... , $\beta_n$ on the diagonal of π΅.
(a) Show that π΄ + π΅ is an upper-triangular matrix with πΌ_1 + π½_1 , ... , πΌΦ_n + π½ _n on the diagonal.
(b) Show that π΄π΅ is an upper-triangular matrix with πΌΡ π½ Ρ , ... , πΌ Φ π½ Φ on the diagonal.
The results in this exercise are used in the proof of 5.81.
SOLUTION
Suppose π΄ and π΅ are π-by-π upper-triangular matrices.
(a) Suppose π, π β {1 , ..., π} with π > π . Then
$(π΄ + π΅)_{j,k}$ = $π΄_{j,k}$ + $B_{j,k}$ = 0 + . = 0.
Thus π΄ + π΅ is an upper-triangular matrix, as desired.
Furthermore,
$(π΄ + π΅)_{j,j}$ = $π΄_{j,j}$+ $B_{j,j}$ = $\alpha_j$+$\large\beta_j$
(b) Suppose π, π β {1 , ..., π} with π > π . Then
(π΄π΅)Φ Λ· Φ =
Φ β
Φ©
= Ρ
π΄Φ Λ· Φ© π΅ Φ© Λ· Φ
=
Φ β Ρ
β
Φ©
= Ρ
π΄Φ Λ· Φ© π΅ Φ© Λ· Φ + Φ β
Φ©
= Φ
π΄Φ Λ· Φ© π΅ Φ© Λ· Φ
=
Φ β Ρ
β
Φ©
= Ρ
0 β
π΅ Φ© Λ· Φ + Φ β
Φ©
= Φ
π΄Φ Λ· Φ© β
0
= 0.
Thus π΄π΅ is an upper-triangular matrix, as desired.
Also,
Edward Frenkel