# 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 $${\mathbb{R}^{4n}}$$.