Home > Algebra, Uncategorized > IMC 2016 – Day 1 – Problem 2

IMC 2016 – Day 1 – Problem 2

Problem 2. Let {k} and {n} be positive integers. A sequence {(A_1,...,A_k)} of {n\times n} matrices is preferred by Ivan the Confessor if {A_i^2 \neq 0} for {1\leq i \leq k}, but {A_iA_j = 0} for {1\leq i,j \leq k} with {i \neq j}. Show that if {k \leq n} in al preferred sequences and give an example of a preferred sequence with {k=n} for each {n}.

Sketch of proof: The fact that {A_iA_j = 0} for {i \neq j} should imply somehow that the images of {A_i}, i.e. the spaces {S_i = \{ y = A_ix, x \in \Bbb{R}^n \}} do not have much in common. Indeed, let’s suppose that {y \in S_i \cap S_j}, then {y = A_i x_i = A_j x_j}. If this happens, then {A_i^2 x_i = A_j^2 x_j = 0}. Now we know that {A_i^2 \neq 0} for every {i} which implies the existence of {x_i} such that {A_i^2 x_i \neq 0} for every {i}.

In this way, the elements {y_i = A_ix_i} are such that {y_i \in S_i} and {y_i \notin S_j} for {i \neq j}. Since {S_i} are linear subspaces of {\Bbb{R}^n} this implies, in particular that the family {\{y_1,...,y_k\}} is linearly independent in {\Bbb{R}^n}. This implies that {k \leq n}.

An example when {k=n} is the family of matrices which have {1} on the {i}-th diagonal position and {0} elsewhere.

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