Notes - Linear Algebra MT23, Cayley-Hamilton theorem

Created: October 26, 2023

Flashcards

Can you state the two equivalent forms of the Cayley-Hamilton theorem?

Suppose

  • $V$ is a finite dimensional vector space
  • $T : V \to V$ linear

Then

\[\chi_T(T) = 0\]

or, alternatively,

\[m_T(x) \mid \chi_T(x)\]

What does the Cayley-Hamilton theorem tell you about the multiplicites of roots of $m _ T$, compared to the multiplicities of roots of $\chi _ T$?
\[\text{multiplicities of } \chi_T \ge \text{multiplicities of } m_T(x)\]

What are the high-level ingredients in the proof of the Cayley-Hamilton theorem, which states that if $V$ is a finite dimensional vector space and $T$ is a linear transformation then $\chi _ T(T) = 0$?
  • Triangular form theorem means we can conjugate into an upper triangular matrix where the diagonal entries are eigenvalues
  • The characteristic polynomial of this matrix is then of the form $\prod (x - \lambda _ i)$
  • This is annihilating polynomial for the upper triangular matirx, so will be for the original matrix

Quickly prove that the index of a nilpotent matrix (i.e. smallest $n$ such that $A^n$ is $0$) must be less than or equal to the dimension of the vector space.

Since $x^k$ annihilates $A$ for some $k$, $m _ T(x) \mid x^k$, and since $\lambda$ is a root of the minimal polynomial iff $\lambda$ is a root of the characteristic polynomial, $\chi _ A(x) = x^n$. But then by the Cayley-Hamilton theorem, $A^n = 0$.

Suppose:

  • $V$ is a finite dimensional vector space over $\mathbb C$
  • $T : V \to V$
  • $U \le V$
  • $U$ is $T$-invariant
  • $\mathcal B$ is a basis of $V$ containing a basis $\mathcal B _ 1$ of $U$ and that $\mathcal B _ 2 := B \setminus B _ 1$.

What useful block decomposition of $T$ is there, and what useful result does this let you deduce?

There exists some matrix $C$ such that

\[{}_ \mathcal{B}[T]_ \mathcal{B} = \left( \begin{array}{cc} {}_ {\mathcal{B}_ 1} [T |_ W]_ {\mathcal{B}_1 } & C \\ 0 & {}_ {q(\mathcal{B}_2)} [\overline T]_ {q(\mathcal{B}_2)} \end{array} \right)\]

This is useful for proving the result that

\[\chi_ T(x) = \chi_ {T|_ W}(x) \chi_ \overline{T}(x)\]

with the special case that if $W = \langle v \rangle$ for some eigenvector

\[\chi_ T(x) = (x - \lambda) \chi_ \overline{T}(x)\]

which can be used in proving the triangular form / Cayley-Hamilton theorem.

Proofs

Prove, via the triangular form theorem, the Cayley-Hamilton theorem, i.e.

$\chi _ T(A) = 0$
  • Place $A$ in triangular form
  • Then the characteristic polynomial has a particularly nice form as a product of (possibly repeated) linear factors
  • Then for arbitrary $v \in \langle e _ 1, \cdots, e _ n\rangle$, consider that $(T - \lambda _ n I)v$ must land in $\langle e _ 1, \cdots, e _ {n-1}\rangle$, and so multiplying all the matrices in $\chi _ T(A)$ together must mean we eventually annihilate each $e _ k$.

There’s also a proof which skips over the triangular form theorem, using a similar argument to the proof of the triangular form theorem but instead the inductive hypothesis is that a matrix is annihilated by its characteristic polynomial.



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