Definitions

The main powerhouse: for \(T:V\to V\) a linear transformation for \(V\in{ \mathsf{Vect}}_k\), map to \(V\in {}_{k[x]}{\mathsf{Mod}}\) by letting polynomials act via \(p(x)\cdot \mathbf{v} \coloneqq p(T)(\mathbf{v})\). Using that \(k[x]\) is a PID iff \(k\) is a field, and we can apply the FTFGMPID to get two decompositions: \begin{align*} V &\cong \bigoplus_{i=1}^n k[x]/ \left\langle{ q_i(x) }\right\rangle && q_{i}(x) \divides q_{i+1}(x) \divides \cdots \\ V &\cong \bigoplus _{j=1}^m k[x] / \left\langle{ p_i(x)^{e_i} }\right\rangle && \text{ with } p_i \text{ not necessarily distinct.} \end{align*}

  • The \(q_i\) are the invariant factors of \(T\)
    • \(q_i\) is the minimal polynomial of \(T\) restricted to \(V_i \coloneqq k[x] / \left\langle{ q_i(x) }\right\rangle\).
    • The largest invariant factor \(q_n\) is the minimal polynomial of \(T\).
    • The product \(\prod_{i=1}^n q_i(x)\) is the characteristic polynomial of \(T\).
  • The \(p_i\) are the elementary divisors of \(T\).
    • Todo: what can you read off of this…?

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A matrix \(A\in \operatorname{Mat}(n\times n; {\mathbf{C}})\) is normal iff \(A^{\dagger} A = AA^{\dagger}\) where \(A^{\dagger}\) is the conjugate transpose.

A matrix \(A\) over \(k\) is semisimple iff \(A\) is diagonalizable over \(k^ \mathsf{Alg}\), the algebraic closure.

A matrix \(A\) over \(k\) is nilpotent iff \(A^k = 0\) for some \(k\geq 1\).

Idea: upper triangular matrices.

A element \(A\) in a ring \(R\) is unipotent iff \(A-1\) is nilpotent.

Idea: an upper-triangular matrix with ones on the diagonal.

Any linear map \(T:V\to V\) over a perfect field decomposes as \(T = S + N\) with \(S\) semisimple (diagonal), \(N\) nilpotent, and \([DN] = 0\). If \(T\) is invertible, then \(T\) decomposes as \(T = SU\) where \(S\) is semisimple, \(U\) is unipotent, and \([UN] = 0\).

\begin{align*} \qty{ \sum W_i}^\perp = \bigcap\qty{W_i^\perp} .\end{align*}

Two matrices \(A,B\) are similar (i.e. \(A = PBP^{-1}\)) \(\iff A,B\) have the same Jordan Canonical Form (JCF).

Two matrices \(A, B\) are equivalent (i.e. \(A = PBQ\)) \(\iff\)

  • They have the same rank,

  • They have the same invariant factors, and

  • They have the same (JCF)

Notation

Some definitions:

  • \(A^t\) is the usual transpose.
  • \(A^{\dagger}\) is the conjugate transpose.
  • A matrix is \(A^{\dagger}\) is adjoint to \(A\) iff \({\left\langle {A\mathbf{x}},~{\mathbf{y}} \right\rangle} = {\left\langle {\mathbf{x}},~{A^{\dagger} \mathbf{y}} \right\rangle}\).
    • \(A\) is self-adjoint iff \(A\) is an adjoint for itself, so \({\left\langle {A\mathbf{x}},~{\mathbf{y}} \right\rangle} = {\left\langle {\mathbf{x}},~{A \mathbf{y}} \right\rangle}\).
  • \(A\) is symmetric iff \(A = A^t\).
    • \(A\) is orthogonal iff \(A^tA = AA^t = I\)
  • \(A\) is Hermitian iff \(A^{\dagger} = A\).
    • \(A\) is normal iff \(AA^{\dagger} = A^{\dagger} A\).
    • \(A\) is unitary iff \(A^{\dagger}A = AA^{\dagger} = I\).
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