The NIM game. A position is a bunch of several rows of matches.
Two players make moves in turn. During his move a player chooses a row
and takes as much matches from this row as he wants. The player taking
last match wins. The problem is to describe positions in which the
first player loose assuming both sides make optimal moves.
Guidelines.
A point moves randomly over the vertices of n-gon.
Each step it moves to one of two neighboring vertices with the probability ½.
Compute the probabilities for the point to be in
different vertices after m steps.
Guidelines.
For a square matrix A describe its ring theoretical centralizer, i.e.
the set of all X such that AX=XA.
Let F be a field of characteristic 0.
Prove that if for some square matrices A and B over F
we have AB-BA=B, then B is nilpotent. Give an example
of matrices over a field of a nonzero characteristic where the above
statement fails.
Hint. Consider first the case of B being a Jordan block.
Let X be a set of pairwise commuting operators
on a finite dimensional vector space V over an algebraically closed field.
Prove that there exists a basis of V such that all elements
of X have upper triangular matrices in this basis.
Guidelines.
Let A and B be n by n complex matrices.
Assume that their ring-theoretical commutator AB-BA has rank 1.
Prove that they have a common eigenvector.
Guidelines.
Let A and X be n by n matrices over an
algebraically closed field.
How many solutions has the matrix equation X²=A?
(Of course, the answer depends on A).
Given an invertible matrix B find the Jordan form, trace and determinant
of the operator
L(X)=B X B –1.
Hint. Choose an appropriate basis (depending on B).
Let A and B be matrices over an algebraically closed field
of size n by n and k by k, respectively.
Assume that they do not have common eigenvalues.
Prove that the matrix equation AX=XB, where X is a
n by k matrix, does not have nonzero solutions.
Hint. Consider the operator L(X)=AX-XB, choose an appropriate basis
(depending on A and B), write the matrix of L in this basis,
and prove that it is nondegenerate.
Let B be a nondegenerate antisymmetric bilinear form on a n-dimensional
vector space V over a field of characteristic not 2.
Prove that there exists a basis v of V such
that the matrix Bv is block-diagonal with diagonal blocks
of the form
Hint. Choose a nonzero vector v1 in V.
Using that B is nondegenerate choose v2 such
that B(v1,v2)=1. Compute the dimension of
B-orthogonal complement to the span of v1 and
v2. Proceed by induction on n.