- In the discussion of Nilpotent transformations, it was asserted that if two n×n matrices
A,B are similar, then A
^{k}is also similar to B^{k}. Why is this so? If two matrices are similar, why must they have the same rank? - If A,B are both invertible, then they are both row equivalent to the identity matrix. Are they necessarily similar? Explain.
- Suppose you have two nilpotent matrices A,B and A
^{k}and B^{k}both have the same rank for all k ≥ 1. Does it follow that A,B are similar? What if it is not known that A,B are nilpotent? Does it follow then? - When we say a polynomial equals zero, we mean that all the coefficients equal 0. If we
assign a different meaning to it which says that a polynomial pequals zero when it is the zero function, ( p= 0 for every λ ∈ F.) does this amount to the same thing? Is there any difference in the two definitions for ordinary fields like ℚ? Hint: Consider for the field of scalars ℤ
_{2}, the integers mod 2 and consider p= λ^{2}+ λ. - Let A ∈ℒwhere V is a finite dimensional vector space with field of scalars F. Let pbe the minimal polynomial and suppose ϕis any nonzero polynomial such that ϕis not one to one and ϕhas smallest possible degree such that ϕis nonzero and not one to one. Show ϕmust divide p.
- Let A ∈ℒwhere V is a finite dimensional vector space with field of scalars F. Let pbe the minimal polynomial and suppose ϕis an irreducible polynomial with the property that ϕx = 0 for some specific x≠0. Show that ϕmust divide p. Hint: First write p= ϕg+ rwhere ris either 0 or has degree smaller than the degree of ϕ. If r= 0 you are done. Suppose it is not 0. Let ηbe the monic polynomial of smallest degree with the property that ηx = 0. Now use the Euclidean algorithm to divide ϕby η. Contradict the irreducibility of ϕ.
- Suppose A is a linear transformation and let the characteristic polynomial be
where the ϕ

_{j}are irreducible. Explain using Corollary 7.3.11 why the irreducible factors of the minimal polynomial are ϕ_{j}and why the minimal polynomial is of the form ∏_{j=1}^{q}ϕ_{j}^{rj}where r_{j}≤ n_{j}. You can use the Cayley Hamilton theorem if you like. - Let
Find the minimal polynomial for A.

- Suppose A is an n × n matrix and let v be a vector. Consider the A cyclic set of vectors
where this is an independent set of vectors but A
^{m}v is a linear combination of the preceding vectors in the list. Show how to obtain a monic polynomial of smallest degree, m, ϕ_{v}such thatNow let

be a basis and let ϕbe the least common multiple of the ϕ_{wk}. Explain why this must be the minimal polynomial of A. Give a reasonably easy algorithm for computing ϕ_{v}. - Here is a matrix.
Using the process of Problem 9 find the minimal polynomial of this matrix. It turns out the characteristic polynomial is λ

^{3}. - Find the minimal polynomial for
by the above technique. Is what you found also the characteristic polynomial?

- Let A be an n × n matrix with field of scalars ℂ. Letting λ be an eigenvalue, show the
dimension of the eigenspace equals the number of Jordan blocks in the Jordan canonical
form which are associated with λ. Recall the eigenspace is ker.
- For any n × n matrix, why is the dimension of the eigenspace always less than or equal to the algebraic multiplicity of the eigenvalue as a root of the characteristic equation? Hint: Note the algebraic multiplicity is the size of the appropriate block in the Jordan form.
- Give an example of two nilpotent matrices which are not similar but have the same minimal polynomial if possible.
- Use the existence of the Jordan canonical form for a linear transformation whose minimal polynomial factors completely to give a proof of the Cayley Hamilton theorem which is valid for any field of scalars. Hint: First assume the minimal polynomial factors completely into linear factors. If this does not happen, consider a splitting field of the minimal polynomial. Then consider the minimal polynomial with respect to this larger field. How will the two minimal polynomials be related? Show the minimal polynomial always divides the characteristic polynomial.
- Here is a matrix. Find its Jordan canonical form by directly finding the eigenvectors and
generalized eigenvectors based on these to find a basis which will yield the Jordan form. The
eigenvalues are 1 and 2.
Why is it typically impossible to find the Jordan canonical form?

- People like to consider the solutions of first order linear systems of equations which are of
the form
where here A is an n × n matrix. From the theorem on the Jordan canonical form, there exist S and S

^{−1}such that A = SJS^{−1}where J is a Jordan form. Define y≡ S^{−1}x. Show y^{′}= Jy. Now suppose Ψis an n × n matrix whose columns are solutions of the above differential equation. ThusNow let Φ be defined by SΦS

^{−1}= Ψ. Show - In the above Problem show that
and so

This is called Abel’s formula and det

is called the Wronskian. Hint: Show it suffices to considerand establish the formula for Φ. Next let

where the ϕ

_{j}are the rows of Φ. Then explain why(9.10) where Φ

_{i}is the same as Φ except the i^{th}row is replaced with ϕ_{i}^{′}instead of the row ϕ_{i}. Now from the form of J,where N has all nonzero entries above the main diagonal. Explain why

Now use this in the formula for the derivative of the Wronskian given in 9.10 and use properties of determinants to obtain

and so the Wronskian detΦ either vanishes identically or never.

- Let A be an n × n matrix and let J be its Jordan canonical form. Recall J is a block
diagonal matrix having blocks J
_{k}down the diagonal. Each of these blocks is of the formNow for ε > 0 given, let the diagonal matrix D

_{ε}be given byShow that D

_{ε}^{−1}J_{k}D_{ε}has the same form as J_{k}but instead of ones down the super diagonal, there is ε down the super diagonal. That is J_{k}is replaced withNow show that for A an n × n matrix, it is similar to one which is just like the Jordan canonical form except instead of the blocks having 1 down the super diagonal, it has ε.

- Let A be in ℒand suppose that A
^{p}x≠0 for some x≠0. Show that A^{p}e_{k}≠0 for some e_{k}∈, a basis for V . If you have a matrix which is nilpotent, (A^{m}= 0 for some m) will it always be possible to find its Jordan form? Describe how to do it if this is the case. Hint: First explain why all the eigenvalues are 0. Then consider the way the Jordan form for nilpotent transformations was constructed in the above. - Suppose A is an n × n matrix and that it has n distinct eigenvalues. How do the minimal polynomial and characteristic polynomials compare? Determine other conditions based on the Jordan Canonical form which will cause the minimal and characteristic polynomials to be different.
- Suppose A is a 3 × 3 matrix and it has at least two distinct eigenvalues. Is it possible that the minimal polynomial is different than the characteristic polynomial?
- If A is an n×n matrix of entries from a field of scalars and if the minimal polynomial of A splits over this field of scalars, does it follow that the characteristic polynomial of A also splits? Explain why or why not.
- Show that if two n × n matrices A,B are similar, then they have the same minimal
polynomial and also that if this minimal polynomial is of the form p= ∏
_{i=1}^{s}ϕ_{i}^{ri}where the ϕ_{i}are irreducible and monic, then kerand kerhave the same dimension. Why is this so? This was what was responsible for the blocks corresponding to an eigenvalue being of the same size. - Show that a given complex n×n matrix is non defective (diagonalizable) if and only if the minimal polynomial has no repeated roots.
- Describe a straight forward way to determine the minimal polynomial of an n × n matrix
using row operations. Next show that if pand p
^{′}are relatively prime, then phas no repeated roots. With the above problem, explain how this gives a way to determine whether a matrix is non defective. - In Theorem 9.3.5 show that each cyclic set β
_{x}is associated with a monic polynomial η_{x}such that η_{x}= 0 and this polynomial has smallest possible degree such that this happens. Show that the cyclic sets β_{xi}can be arranged such that η_{xi+1}∕η_{xi}. - Show that if A is a complex n × n matrix, then A and A
^{T}are similar. Hint: Consider a Jordan block. Note that - Let A be a linear transformation defined on a finite dimensional vector space V . Let the
minimal polynomial be ∏
_{i=1}^{q}ϕ_{i}^{mi}and letbe the cyclic sets such thatis a basis for ker. Let v = ∑_{i}∑_{j}v_{j}^{i}. Now let qbe any polynomial and suppose thatShow that it follows q

= 0 . Hint: First consider the special case where a basis for V isand qx = 0.

Download PDFView PDF