Let . Prove that

Solution

The LHS is equal to which by AM – GM is less or equal to

where . Since it follows from Bernoulli inequality that .

No invertible matrices

Show that there do not exist invertible matrices such that and .

Solution

Suppose, on the contrary, that such matrices do exist. Then

and also

Using the fact that we deduce that

The last means that which is impossible because both and are invertible ( and so must be the product ). Hence, the conclusion follows.

Matrix determinant inequality

Suppose that all eigenvalues of are positive real numbers. Show that

Solution

Let the eigenvalues of be , . Consider the Jordan normal form of ;  this Jordan form is an upper triangular matrix that has the eigenvalues of in the main diagonal. Let this matrix be called .  Furthermore , as a matrix and its Jordan normal form are similar. As is upper triangular, its inverse is given by an upper triangular matrix whose diagonal entries are the inverses of the diagonal entries of . That is,

is an upper-triangular matrix, and its determinant can be computed simply as the product of the elements of the diagonal. Hence,

due to the inequality for all .
Note: The main point here is that if is an eigenvalue of then is an eigenvalue of . Taking into account that

yields a shorter solution to the problem. No need for upper triangular matrices.

Root inequality

Let be positive real numbers such that . Prove that

Solution

Well if we apply AM-GM to we obtain

(1)

and similarly if we apply AM – GM to we obtain

(2)

We have successively,

Group homomorphism of uncountable kernel

Let be a group homomorphism. We proved in this question that its kernel is infinite. We are proving now that it is also uncountable.

Solution

Before we proceed with the proof we are stating that not all homomorphisms are of the form . We can find non trivial homomorphisms. But all continuous are of the above form.

Let be the group of roots of unity. Both groups and are uniquely divisible and thus are vector spaces over .

Since the positive reals are closed under multiplication, it’s easy to see that

Using the axiom of choice  ,  we   construct a group homomorphism

Hence the kernel is uncountable.

Note: is  a direct summand of . That is because

( since is injective )  and so the exact sequence

splits.