Linear algebra

Here we recall some basics of linear algebra, which is assumed to be familiar to the reader. We start by collecting some terminology:

  • \({A^{\textrm{T}}}\): transpose of \({A}\), reflecting entries across the diagonal
  • \({A^{\textrm{T}}=-A}\): anti-symmetric (AKA skew-symmetric) matrix
  • \({A^{\dagger}}\): adjoint (AKA hermitian conjugate, conjugate transpose) of a matrix, the transposed complex conjugate (also denoted \({A^{*}}\))
  • \({A^{\dagger}=A}\): hermitian matrix, a matrix that is self-adjoint
  • \({A^{\dagger}=-A}\): anti-hermitian (AKA skew-hermitian) matrix \({\Rightarrow iA}\) is hermitian
  • \({A^{\dagger}A=I}\): unitary (orthogonal) matrix for complex (real) entries
  • \({A^{\dagger}A=AA^{\dagger}}\): normal matrix, e.g. a hermitian or unitary matrix
  • Eigenvalues: scalars \({a}\) such that \({Av=av}\) for vectors \({v}\), the eigenvectors
  • \({\textrm{tr}(A)}\): the trace of the matrix \({A}\), the sum of the diagonal entries
  • \({\textrm{det}(A)}\): determinant of \({A}\)
  • Singular means \({\textrm{det}(A)=0}\), unimodular can mean either \({\left|\textrm{det}(A)\right|=1}\) or \({\textrm{det}(A)=1}\)
  • Similarity transformation: \({A\rightarrow BAB^{-1}}\) by a nonsingular matrix \({B}\)

In terms of the underlying abstract linear transformation, the adjoint is defined by \({\left\langle v,Aw\right\rangle =\left\langle A^{\dagger}v,w\right\rangle}\), while the trace is the contraction of the associated type \({\left(1,1\right)}\) tensor and, recalling its properties, \({\textrm{det}(A)}\) is the volume change associated with \({A}\) applied to an orthonormal basis.

Some basic facts are:

  • A similarity transformation \({A\rightarrow BAB^{-1}}\) is equivalent to a change of the basis defining the vector components operated on by \({A}\), where the change of basis has matrix \({B^{-1}}\) so that \({v\rightarrow Bv}\)
  • The eigenvalues, determinant and trace of \({A}\) are independent of basis \({\Rightarrow}\) unchanged by a similarity transformation
  • The trace is a cyclic linear map: \({\textrm{tr}(ABC)=\textrm{tr}(BCA)=\textrm{tr}(CAB)}\)
  • The determinant is a multiplicative map: \({\textrm{det}(rAB)=r^{n}\textrm{det}(A)\textrm{det}(B)}\)
  • The trace equals the sum of eigenvalues; the determinant equals their product
  • \({\textrm{det}\left(\textrm{exp}(A)\right)=\textrm{exp}\left(\textrm{tr}(A)\right)}\); \({\left(\textrm{exp}(A)\right)^{\dagger}=\textrm{exp}(A^{\dagger})}\)
  • \({\textrm{det}(I+\varepsilon A)=1+\varepsilon \textrm{tr}(A)+\dots}\)
  • A hermitian matrix has real eigenvalues and orthogonal eigenvectors
  • The tensor product of hermitian / unitary matrices is hermitian / unitary
  • Diagonalizable matrices commute iff they are simultaneously diagonalizable
  • Spectral theorem: a matrix is normal iff it can be diagonalized by a unitary similarity transformation; a real matrix is symmetric iff it can be diagonalized by an orthogonal similarity transformation

As previously noted, we can geometrically interpret an element of a matrix group with real entries as a transformation on \({\mathbb{R}^{n}}\). Such a transformation preserves the orientation of \({\mathbb{R}^{n}}\) if its determinant is positive, and preserves volumes if the determinant has absolute value one.

Any bilinear form \({\varphi}\) on \({\mathbb{R}^{n}}\) can be represented by a matrix in the standard basis, with the form operation then being \({\varphi(v,w)=v^{\textrm{T}}\varphi w}\). The group of matrices that preserve a form \({\varphi}\) consists of matrices \({A}\) that satisfy \({\varphi\left(Av,Aw\right)=\varphi\left(v,w\right)\Leftrightarrow\left(Av\right)^{\textrm{T}}\varphi(Aw)=v^{\textrm{T}}\varphi w\Leftrightarrow A^{\textrm{T}}\varphi A=\varphi}\). Any similarity transformation simply changes the basis of each \({A}\), leaving the group of matrices that preserve the form unchanged. Thus we can concern ourselves only with a canonical form of the preserved form. In \({\mathbb{R}^{n}}\), we have several naturally defined forms:

  • The Euclidean inner product, with canonical form \({I}\)
  • The pseudo-Euclidean inner product of signature \({(r,s)}\), with canonical form \({\left(r+s=n\right)}\)

    \(\displaystyle \eta=\begin{pmatrix}I_{r} & 0\\ 0 & -I_{s} \end{pmatrix} \)

  • The symplectic form, with canonical form

    \(\displaystyle J=\begin{pmatrix}0 & I_{n/2}\\ -I_{n/2} & 0 \end{pmatrix} \)

Any matrix group defined as preserving one of these canonical forms then preserves all forms in the corresponding similarity class. Some matrix groups with entries in \({\mathbb{C}}\) or \({\mathbb{H}}\) can also be viewed as preserving a form in the vector space \({\mathbb{C}^{n}}\) or module \({\mathbb{H}^{n}}\), but we will mainly view these as linear transformations on \({\mathbb{R}^{2n}}\) or \({\mathbb{R}^{4n}}\).

An Illustrated Handbook