Linear algebra

Here we recall some basics of linear algebra, which is assumed to be familiar to the reader. We start by collecting some matrix 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 \({B}\)

As previously noted, we can geometrically interpret an element of a matrix group with real entries as a linear 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. In terms this transformation, the matrix rank is the dimension of its image, and the adjoint is defined by \({\left\langle v,Aw\right\rangle =\left\langle A^{\dagger}v,w\right\rangle}\).

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 \({A}\) as a linear transformation, 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

Recalling the section on combinatorial notations, \({\textrm{det}(A)}\) is the volume change multiple associated with \({A}\) applied to an orthonormal basis, so that \({\mathrm{tr}(A)}\) is then the volume change addition per unit \({t}\) of \({\mathrm{exp}(tA)}\).

Any bilinear form \({\varphi}\) on \({\mathbb{R}^{n}}\) can be also 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; the matrix representation of the form in the old basis becomes \({\left(B^{-1}v\right)^{\textrm{T}}\left(B^{\textrm{T}}\varphi B\right)\left(B^{-1}w\right)=v^{\textrm{T}}\varphi w}\), where the matrices \({B^{\textrm{T}}\varphi B}\) and \({\varphi}\) are called congruent. We can therefore disregard congruences and 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