- 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? - (Review problem.) 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 minimum 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 minimum 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.
- Let
Find the minimum 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 minimum polynomial of A. Give a reasonably easy algorithm for computing ϕ_{v}. - Here is a matrix.
Using the process of Problem 8 find the minimum polynomial of this matrix. Determine whether it can be diagonalized from its minimum polynomial.

- Let A be an n × n matrix with field of scalars ℂ or more generally, the minimum polynomial splits.
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 minimum polynomial if possible.
- 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?

- Let A be an n × n matrix and let J be its Jordan canonical form. Here F = ℝ or ℂ. 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. - Show that if two n×n matrices A,B are similar, then they have the same minimum polynomial and
also that if this minimum 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. - In Theorem 7.1.6 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 - (Extra important) Let A be an n×n matrix. The trace of A,traceis defined as ∑
_{i}A_{ii}. It is just the sum of the entries on the main diagonal. Show trace= trace. Suppose A is m×n and B is n × m. Show that trace= trace. Now show that if A and B are similar n × n matrices, then trace= trace. Recall that A is similar to B means A = S^{−1}BS for some matrix S. - (Extra important) If A is an n × n matrix and the minimum polynomial splits in F the field of
scalars, show that traceequals the sum of the eigenvalues listed according to multiplicity according to number of times they occur in the Jordan form. Next, show that this is true even if the minimum polynomial does not split.
- Let A be a linear transformation defined on a finite dimensional vector space V . Let the minimum
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.

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