In the first lecture, we discussed some examples of groups and some basics of linear algebra.

Groups

• the elements of the cylic group \(\mathbb{Z}/n\mathbb{Z}\) are the equivalence classes of integers under the relation “\(\equiv \pmod{n}\)”

  this group is additive

• we observed that the mapping \(\phi:\mathbb{R} \to \mathbf{S}^1\) given by \(\phi(t) = e^{2\pi i t}\) is a group homomorphism since \(\phi(t + s) = \phi(t) \phi(s)\) for all \(t,s \in \mathbb{R}\).

  we observed that \(\ker \phi = \mathbb{Z}\), and that - by the First Isomorphism Theorem - \(\phi\) induces an isomorphism \[\overline{\phi}: \mathbb{R}/\mathbb{Z} \to \mathbf{S}^1.\]

• for a non-zero natural number the symmetric group \(S_n\) is the collection of all bijections \(I_n \to I_n\) where \(I_n = \{1,2,\cdots,n\}\).

  We may sometimes use cycle notation for elements of \(S_n\).

  The subgroup \[H=\langle (1234), (14)(23) \rangle\] has order \(8\) and is sometimes called the dihedral group \(D_4\) or \(D_8\) – it has order 8.

• Let \(F\) be a field.

  Recall that typically examples are: \(F = \mathbb{R},\mathbb{C},\mathbb{Q},\mathbb{Z}/p\mathbb{Z}\) for a prime number \(p\).

  The set \[\operatorname{GL}_n(F) = \{\text{all invertible $n \times n$ matrices with entries in $F$}\}\] forms a group under matrix multiplication.

  The determinant function yields a group homomorphism \[\det:\operatorname{GL}_n(F) \to F^\times\] (here \(F^\times\) means \(F \setminus \{0\}\), which is a commutative group under multiplication in the field \(F\)).

Linear Algebra

Let \(F\) be a field. An \(F\)-vector space \(V\) is an additive abelian group together with an operation of scalar multiplication – this amounts to a function \[F \times V \to V\] – satisfying certain axioms.

If \(V,W\) are \(F\)-vector spaces, a linear mapping \(T:V \to W\) is a function which satisfies \[T(\alpha v + w) = \alpha T(v) + T(w).\]

Let’s suppose that \(V\) is finite dimensional and that \(\phi:V \to V\).

We write \(\phi^2 = \phi \circ \phi\) and more generally \(\phi^n = \phi \circ \phi^{n-1}\).

trace, det, char poly

The trace of a matrix \(M=[M_{ij}]\) is the sum of the diagonal entries: \[\operatorname{tr}(M) = \sum_{i=1}^n M_{ii}.\]

I’m assuming you recall the definition of the determinant \(\det M\).

The characteristic polynomial \(\operatorname{cp}_M(X) \in F[X]\) of \(M\) is defined to be \[\operatorname{cp}_M(X) = \det(M - X \cdot \mathbf{I}_n).\]

For a linear transformation \(\phi\) we define

• \(\operatorname{tr}(\phi) = \operatorname{tr}([\phi]_{\mathcal{B}})\)

• \(\operatorname{det}(\phi) = \operatorname{det}([\phi]_{\mathcal{B}})\)

• \(\operatorname{cp}_\phi(X) = \operatorname{cp}_{[\phi]_{\mathcal{B}}}(X)\)

Proposition

\(\operatorname{tr}(\phi)\), \(\operatorname{det}(\phi)\), and \(\operatorname{p}_\phi(X)\) are independent of the choice \(\mathcal{B}\) of basis for \(V\).

The main point here is that if \(\mathcal{B}\) and \(\mathcal{B}'\) are two basis for \(V\), there is an invertible matrix (“change of basis matrix”) \(P\) for which \[[\phi]_{\mathcal{B}} = P [\phi]_{\mathcal{B}'} P^{-1}.\]

Evaluation of polynomials at linear transformations

Suppose that \(f = f(X) \in F[X]\) is a polynomial; thus \[f = \sum_{i=0}^N a_i X^i\] for some coefficients \(a_i \in F\).

We may evaluate the polynomial \(f\) at the linear endmorphism \(\phi\):

\[f(\phi) = \sum_{i=0}^N a_i \phi^i.\]

Proposition

Let \(\phi:V \to V\) be a linear transformation, and let \[I = \{f \in F[X] \mid f(\phi) = 0\}.\] Then \(I\) is an ideal in the polynomial ring \(F[X]\). In particular, there is a unique monic polynomial \(m_\phi(X) \in F[x]\) for which \(I = m_\phi(X) F[X]\).

In particular, if \(f \in F[X]\) and \(f(\phi)=0\), then \(m_\phi \vert f\).

Theorem (Cayley-Hamilton)

Let \(\phi:V \to V\) be a linear transformation, and let \(\operatorname{cp}(X) = \operatorname{cp}_\phi(X) \in F[X]\) be the characteristic polynomial.

Then \(\operatorname{cp}(\phi) = 0\).

Recall that the eigenvalues of \(\phi\) are precisely the roots of the characteristic polynomial. The Cayley-Hamilton Theorem implies that any root of the minimal polynomial is an eigenvalue. In fact, we have the converse as well:

Proposition:

If \(\lambda \in F\) is an eigenvalue of \(\phi\) – i.e. a root of the characteristic polynomial – then \(\lambda\) is a root of the minimal polynomial.

Theorem:

\(\phi\) is diagonalizable – i.e. \(V\) has a basis of eigenvectors for \(\phi\) – if and only the minimal polynomial has no multiple roots.

Remark:

This theorem should be proved in Math215-216 here at Tufts using the Fundamental Theorem for modules over a PID. We don’t need the full force of this result in our class.

Example

Suppose that \(\phi:V \to V\) satisfies \(\phi^N = \operatorname{id}_V\) for some positive natural number \(N\).

We suppose that \(F\) is algebraically closed and of characteristic zero.

Notice that the polynomial \(f(X) = X^N -1 \in F[X]\) has distinct roots.

(If \(F = \mathbb{C}\), these roots are exactly \[\{\exp(2\pi k i/n) \mid 0 \le k < N\}.\] )

Since the minimal polynomial of \(\phi\) divides \(f\), we see that the minimal polynomial has distinct roots and hence \(\phi\) is diagonalizable by the Theorem quoted above.