MTH 525, FALL 2006

Brief Notes for Week 10

(very tentative)

 

Monday, Oct 30

           We finish section 6.2.

 

Wednesday, November 1, 2006

The annihilating polynomials of a matrix.

 

The minimal polynomial of a matrix.

 

Theorem 3.  (p. 193.)

           Let T a linear operator with characteristic polynomial f  and minimal polynomial p.  The polynomials f and p have the same roots.

 

           Proof. 

           First, suppose p(c) = 0, so that c is a root of p.  We want to show that  c  is an eigenvalue for  T  by displaying an eigenvector for c.

           Let us assume the degree of p is greater than 1.  (Why?  What if the degree of p is one or less?)

           Write p = (x-c)q where q is a polynomial of smaller degree.  Then q(T) is not zero.  (Why?) 

           Since q(T) is not the zero transformation, there is a vector  a  such that  b = q(T)a  is not zero.  (Why?)

           Then b  is an eigenvector corresponding to c.  (Verify.)

           On the other hand, letÕs assume  f(c) = 0 (so c is an eigenvalue of T) and  a is a nonzero vector such that  Ta = ca.  Now q(T)a = q(c)a.  (Why?) 

           But since q(T) = 0, we have that q(c)a=0.  Since q(c) is a scalar and a is a nonzero vector, this implies that q(c) = 0.  So the eigenvalue c is a root of the minimal polynomial.

 

           Do p. 197: 1-7.  (IÕll do 1, 2, 7.)

 

           Cayley Hamilton:  the minimal polynomial divides the characteristic polynomial.

 

           Invariant subspaces.   What this says about the similarity class of T. 

           There are always some trivial T-invariant subspacesÉ.  And eigenspaces are T-invariant.

 

The Lemma on p. 200:  Suppose W is an invariant subspace of T and let U be the restriction of T to W.  Then the characteristic polynomial of U divides the characteristic polynomial of T; the minimal polynomial of U divides the minimal polynomial of T.

           Proof.  Take a basis of W and extend it to a basis of V.  Now write out the matrix A  which represents the linear transformation T with respect to that basis.  (It will have a big zero matrix in the lower left!)

           Look at the determinant of (A-xI).

           Look at all powers of A and so the minimal polynomial of B must divide AÉ.

 

Given a T-invariant subspace W.  The T-conductor (ST(a;W)) of a into W is the set (ideal) of polynomials p such that p(T)a is in W.

 

The Lemma on p. 202.

           Suppose the minimal polynomial p for T has only linear factors.  Suppose also that W is a T-invariant subspace.  Then there exists a vector a such that a is not in W but (T-cI)a is in W for some eigenvalue c of T.

           Proof.  Suppose b is not in W and p = (x-c1)^d1(x-c2)^d2 É etc.  Let g be the ÒstufferÓ (conductor) for b. 

           Now g divides p (Why?).  And g =(x-cj)h where h is not a ÒconductorÓ so hb is not in W.  Set a = hb.

          

Theorem 5  (p. 203, a corollary to Lemma.)

           If the min poly of T factors completely then T is similar to a triangular matrix (and so is triangulable.) 

 

Proof.  We construct a basis for V inductively, using the previous Lemma.

 

Corollary.  If F is algebraically closed the T is triangulable.

 

Theorem 6 (p. 204)

           T is diagonalizable if and only if the min poly factors into distinct linear terms.

           Proof.  One direction is obvious. (Suppose T is diagonalizableÉ.)

           Suppose W is the subspace spanned by the eigenvectors of T.  This is a T-invariant space.  If W is not all of V then there is a vector a not in W such that b:=(T-cI)a lies in W and so b is a sum of eigenvectors, b = b1+b2+Ébk.

           Pick any polynomial h; h(T)b = h(c1)b1 + h(c2)b2 + É h(ck)bk.

           Factor p as (x-c1)(x-c2)É(x-ck)  (Assumption)

           Note that if p = (x-c)q then q-q(c) =(x-c)h for some polynomial h.

           Apply  this to a:  q(T)a-q(c)a = h(T)(T-cI)a = h(T)b which is in W. 

 

           But 0=p(T)a=(T-cI)q(T)a and so q(T)a must be in W.  But a is not, so q(c) must be zero. 

 

P 205: 1-4 (IÕll do 2, 3.)

 

Wednesday, October 25

           We continue in chapter 6.

           We develop the Cayley-Hamilton theorem (in section 6.3)

           We introduce invariant subspaces (section 6.4)

           Do p. 205.

           We discuss triangulation, diagonalization, beginning with section 6.5.

           We finish chapter 6, covering invariant direct sums (6.7) and the primary decomposition theorem (6.8).

 

           Assignment 7 is collected.

           We study triangulation, diagonalization (section 6.5) of linear transformations.

           Do p. 208.

           We study direct-sum decompositions (section 6.6) of linear transformations.

           Do p. 213.

 

 

Last modified October 1, 2006