If
the group is called Abelian or commutative. Vector spaces form an Abelian group for addition and multiplication: $1\cdot\vec{a}=\vec{a}$, $\lambda(\mu\vec{a})=(\lambda\mu)\vec{a}$, $(\lambda+\mu)(\vec{a}+\vec{b})=\lambda\vec{a}+\lambda\vec{b}+\mu\vec{a}+\mu\vec{b}$.
$W$ is a linear subspace if $\forall \vec{w}_1,\vec{w}_2\in W$ holds: $\lambda\vec{w}_1+\mu\vec{w}_2\in W$.
$W$ is an invariant subspace of $V$ for the operator $A$ if $\forall\vec{w}\in W$ holds: $A\vec{w}\in W$.
The set vectors $\{\vec{a}_n\}$ is linear independent if: \[ \sum\limits_i\lambda_i\vec{a}_i=0~~\Leftrightarrow~~\forall_i\lambda_i=0 \] The set $\{\vec{a}_n\}$ is a basis if it is 1. independent and 2. $V=<\vec{a}_1,\vec{a_2},...>=\sum\lambda_i\vec{a}_i$.
The transpose of $A$ is defined by: $a_{ij}^T=a_{ji}$. For this holds $(AB)^T=B^TA^T$ and $(A^T)^{-1}=(A^{-1})^T$. For the inverse matrix holds: $(A\cdot B)^{-1}=B^{-1}\cdot A^{-1}$. The inverse matrix $A^{-1}$ has the property that $A\cdot A^{-1}=I\hspace{-1mm}I$ and can be found by diagonalization: $(A_{ij}|I\hspace{-1mm}I)\sim(I\hspace{-1mm}I |A_{ij}^{-1})$.
The inverse of a $2\times2$ matrix is: \[ \left(\begin{array}{cc}a&b\\ c&d\end{array}\right)^{-1}=\frac{1}{ad-bc} \left(\begin{array}{cc}d&-b\\ -c&a\end{array}\right) \]
The determinant function $D=\det(A)$ is defined by: \[ \det(A)=D(\vec{a}_{*1},\vec{a}_{*2},...,\vec{a}_{*n}) \] For the determinant $\det(A)$ of a matrix $A$ holds: $\det(AB)=\det(A)\cdot\det(B)$. Een $2\times2$ matrix has determinant: \[ \det\left(\begin{array}{cc}a&b\\ c&d \end{array}\right)=ad-cb \] The derivative of a matrix is a matrix with the derivatives of the coefficients: \[ \frac{dA}{dt}=\frac{da_{ij}}{dt}~~~\mbox{and}~~~\frac{dAB}{dt}=B\frac{dA}{dt}+A\frac{dB}{dt} \] The derivative of the determinant is given by: \[ \frac{d\det(A)}{dt}=D(\frac{d\vec{a}_1}{dt},...,\vec{a}_n)+ D(\vec{a}_1,\frac{d\vec{a}_2}{dt},...,\vec{a}_n)+...+D(\vec{a}_1,...,\frac{d\vec{a}_n}{dt}) \] When the rows of a matrix are considered as vectors the row rank of a matrix is the number of independent vectors in this set. Similar for the column rank. The row rank equals the column rank for each matrix.
Let $\tilde{A}:\tilde{V}\rightarrow\tilde{V}$ be the complex extension of the real linear operator $A:V\rightarrow V$ in a finite dimensional $V$. Then $A$ and $\tilde{A}$ have the same caracteristic equation.
When $A_{ij}\in I\hspace{-1mm}R$ and $\vec{v}_1+i\vec{v_2}$ is an eigenvector of $A$ at eigenvalue $\lambda=\lambda_1+i\lambda_2$, than holds:
If $\vec{k}_n$ are the columns of $A$, than the transformed space of $A$ is given by:
\[
R(A)=
The equation
\[
A\cdot\vec{x}=\vec{0}
\]
has exactly one solution $\neq\vec{0}$ if $\det(A)=0$, and if
$\det(A)\neq0$ the solution is $\vec{0}$.
Cramer's rule for the solution of systems of linear equations is: let the
system be written as
\[
A\cdot\vec{x}=\vec{b}\equiv\vec{a}_1x_1+...+\vec{a}_nx_n=\vec{b}
\]
then $x_j$ is given by:
\[
x_j=\frac{D(\vec{a}_1,...,\vec{a}_{j-1},\vec{b},\vec{a}_{j+1},...,\vec{a}_n)}{\det(A)}
\]
Some common linear transformations are:
For a projection holds: $\vec{x}-P_W(\vec{x})\perp P_W(\vec{x})$ and $P_W(\vec{x})\in W$.
If for a transformation $A$ holds: $(A\vec{x},\vec{y})=(\vec{x},A\vec{y})=(A\vec{x},A\vec{y})$,
than $A$ is a projection.
Let $A:W\rightarrow W$ define a linear transformation; we define:
Than $A(S)$ is a linear subspace of $W$ and the {\it inverse transformation}
$A^\leftarrow(T)$ is a linear subspace of $V$. From this follows that $A(V)$ is
the {\it image space} of $A$, notation: ${\cal R}(A)$. $A^\leftarrow(\vec{0})=E_0$
is a linear subspace of $V$, the {\it null space} of $A$, notation:
${\cal N}(A)$. Then the following holds:
\[
{\rm dim}({\cal N}(A))+{\rm dim}({\cal R}(A))={\rm dim}(V)
\]
The distance $d$ between 2 points $\vec{p}$ and $\vec{q}$ is given by
$d(\vec{p},\vec{q})=\|\vec{p}-\vec{q}\|$.
In $I\hspace{-1mm}R^2$ holds:
The distance of a point $\vec{p}$ to the line $(\vec{a},\vec{x})+b=0$ is
\[
d(\vec{p},\ell)=\frac{|(\vec{a},\vec{p})+b|}{|\vec{a}|}
\]
Similarly in $I\hspace{-1mm}R^3$:
The distance of a point $\vec{p}$ to the plane $(\vec{a},\vec{x})+k=0$ is
\[
d(\vec{p},V)=\frac{|(\vec{a},\vec{p})+k|}{|\vec{a}|}
\]
This can be generalized for $I\hspace{-1mm}R^n$ and $\mathbb{C}^n$ (theorem from Hesse).
The matrix $A_\alpha^\beta$ transforms a vector given w.r.t. a basis
$\alpha$ into a vector w.r.t. a basis $\beta$. It is given by:
\[
A_\alpha^\beta=\left(\beta(A\vec{a}_1),...,\beta(A\vec{a}_n)\right)
\]
where $\beta(\vec{x})$ is the representation of the vector $\vec{x}$
w.r.t. basis $\beta$.
The transformation matrix $S_\alpha^\beta$ transforms vectors from
coordinate system $\alpha$ into coordinate system $\beta$:
\[
S_\alpha^\beta:=I\hspace{-1mm}I_\alpha^\beta=\left(\beta(\vec{a}_1),...,\beta(\vec{a}_n)\right)
\]
and $S_\alpha^\beta\cdot S_\beta^\alpha=I\hspace{-1mm}I$
The matrix of a transformation $A$ is than given by:
\[
A_\alpha^\beta=\left(A_\alpha^\beta\vec{e}_1,...,A_\alpha^\beta\vec{e}_n\right)
\]
For the transformation of matrix operators to another coordinate system holds:
$A_\alpha^\delta=S_\lambda^\delta A_\beta^\lambda S_\alpha^\beta$,
$A_\alpha^\alpha=S_\beta^\alpha A_\beta^\beta S_\alpha^\beta$ and
$(AB)_\alpha^\lambda=A_\beta^\lambda B_\alpha^\beta$.
Further is $A_\alpha^\beta=S_\alpha^\beta A_\alpha^\alpha$,
$A_\beta^\alpha=A_\alpha^\alpha S_\beta^\alpha$. A vector is transformed via
$X_\alpha=S_\alpha^\beta X_\beta$.
The eigen values $\lambda_i$ are independent of the chosen basis.
The matrix of $A$ in a basis of eigenvectors, with $S$ the transformation matrix
to this basis, $S=(E_{\lambda_1},...,E_{\lambda_n})$, is given by:
\[
\Lambda=S^{-1}AS={\rm diag}(\lambda_1,...,\lambda_n)
\]
When 0 is an eigen value of $A$ than $E_0(A)={\cal N}(A)$.
When $\lambda$ is an eigen value of $A$ holds: $A^n\vec{x}=\lambda^n\vec{x}$.
When $W$ is an invariant subspace if the isometric transformation $A$ with
dim$(A)<\infty$, than also $W^\perp$ is an invariante subspace.
Let $A:V\rightarrow V$ be orthogonal with dim$(V)<\infty$. Than $A$ is:
Direct orthogonal if $\det(A)=+1$. $A$ describes a rotation.
A rotation in $I\hspace{-1mm}R^2$ through angle $\varphi$ is given by:
\[
R=
\left(\begin{array}{cc}
\cos(\varphi)&-\sin(\varphi)\\
\sin(\varphi)&\cos(\varphi)
\end{array}\right)
\]
So the rotation angle $\varphi$ is determined by Tr$(A)=2\cos(\varphi)$
with $0\leq\varphi\leq\pi$. Let $\lambda_1$ and $\lambda_2$ be the roots of
the characteristic equation, than also holds:
$\Re(\lambda_1)=\Re(\lambda_2)=\cos(\varphi)$, and $\lambda_1=\exp(i\varphi)$,
$\lambda_2=\exp(-i\varphi)$.
In $I\hspace{-1mm}R^3$ holds: $\lambda_1=1$, $\lambda_2=\lambda_3^*=\exp(i\varphi)$. A
rotation over $E_{\lambda_1}$ is given by the matrix
\[
\left(\begin{array}{ccc}
1&0&0\\
0&\cos(\varphi)&-\sin(\varphi)\\
0&\sin(\varphi)&\cos(\varphi)
\end{array}\right)
\]
Mirrored orthogonal if $\det(A)=-1$. Vectors from $E_{-1}$ are mirrored
by $A$ w.r.t. the invariant subspace $E^\perp_{-1}$. A mirroring in $I\hspace{-1mm}R^2$
in $<(\cos(\frac{1}{2}\varphi),\sin(\frac{1}{2}\varphi))>$ is given by:
\[
S=
\left(\begin{array}{cc}
\cos(\varphi)&\sin(\varphi)\\
\sin(\varphi)&-\cos(\varphi)
\end{array}\right)
\]
Mirrored orthogonal transformations in $I\hspace{-1mm}R^3$ are rotational mirrorings:
rotations of axis $<\vec{a}_1>$ through angle $\varphi$ and mirror plane
$<\vec{a}_1>^\perp$. The matrix of such a transformation is given by:
\[
\left(\begin{array}{ccc}
-1&0&0\\
0&\cos(\varphi)&-\sin(\varphi)\\
0&\sin(\varphi)&\cos(\varphi)
\end{array}\right)
\]
For all orthogonal transformations $O$ in $I\hspace{-1mm}R^3$ holds that
$O(\vec{x})\times O(\vec{y})=O(\vec{x}\times\vec{y})$.
$I\hspace{-1mm}R^n$ $(n<\infty)$ can be decomposed in invariant subspaces with dimension
1 or 2 for each orthogonal transformation.
Theorem: for a $n\times n$ matrix $A$ the following statements are
equivalent:
For each matrix $B\in I\hspace{-1mm}M^{m\times n}$ holds: $B^TB$ is symmetric.
If the transformations $A$ and $B$ are Hermitian, than their product $AB$ is
Hermitian if: $[A,B]=AB-BA=0$. $[A,B]$ is called the commutator of $A$
and $B$.
The eigenvalues of a Hermitian transformation belong to $I\hspace{-1mm}R$.
A matrix representation can be coupled with a Hermitian operator $L$.
W.r.t. a basis $\vec{e}_i$ it is given by $L_{mn}=(\vec{e}_m,L\vec{e}_n)$.
Definition: the linear transformation $A$ is normal in a complex
vector space $V$ if $A^*A=AA^*$. This is only the case if for its matrix $S$
w.r.t. an orthonormal basis holds: $A^\dagger A=AA^\dagger$.
If $A$ is normal holds:
Let the different roots of the characteristic equation of $A$ be $\beta_i$ with
multiplicities $n_i$. Than the dimension of each eigenspace $V_i$ equals
$n_i$. These eigenspaces are mutually perpendicular and each vector
$\vec{x}\in V$ can be written in exactly one way as
\[
\vec{x}=\sum_i\vec{x}_i~~~\mbox{with}~~~\vec{x}_i\in V_i
\]
This can also be written as: $\vec{x}_i=P_i\vec{x}$ where $P_i$ is a projection
on $V_i$. This leads to the {\it spectral mapping theorem}: let $A$ be a normal
transformation in a complex vector space $V$ with dim$(V)=n$. Than:
Lemma: if $E_\lambda$ is the eigenspace for eigenvalue $\lambda$ from
$A_1$, than $E_\lambda$ is an invariant subspace of all transformations
$A_i$. This means that if $\vec{x}\in E_\lambda$, than $A_i\vec{x}\in E_\lambda$.
Theorem. Consider $m$ commuting Hermitian matrices $A_i$. Than there
exists a unitary matrix $U$ so that all matrices $U^\dagger A_iU$ are diagonal.
The columns of $U$ are the common eigenvectors of all matrices $A_j$.
If all eigenvalues of a Hermitian linear transformation in a $n$-dimensional
complex vector space differ, than the normalized eigenvector is known except
for a phase factor $\exp(i\alpha)$.
Definition: a commuting set Hermitian transformations is called
complete if for each set of two common eigenvectors $\vec{v}_i,\vec{v}_j$
there exists a transformation $A_k$ so that $\vec{v}_i$ and $\vec{v}_j$ are
eigenvectors with different eigenvalues of $A_k$.
Usually a commuting set is taken as small as possible. In quantum physics one
speaks of commuting observables. The required number of commuting observables
equals the number of quantum numbers required to characterize a state.
Due to (1) holds: $(\vec{a},\vec{a})\in I\hspace{-1mm}R$. The inner product space $\mathbb{C}^n$ is
the complex vector space on which a complex inner product is defined by:
\[
(\vec{a},\vec{b})=\sum_{i=1}^na_i^*b_i
\]
For function spaces holds:
\[
(f,g)=\int\limits_a^bf^*(t)g(t)dt
\]
For each $\vec{a}$ the length $\|\vec{a}\|$ is defined by:
$\|\vec{a}\|=\sqrt{(\vec{a},\vec{a})}$. The following holds:
$\|\vec{a}\|-\|\vec{b}\|\leq\|\vec{a}+\vec{b}\|\leq\|\vec{a}\|+\|\vec{b}\|$,
and with $\varphi$ the angle between $\vec{a}$ and $\vec{b}$ holds:
$(\vec{a},\vec{b})=\|\vec{a}\|\cdot\|\vec{b}\|\cos(\varphi)$.
Let $\{\vec{a}_1,...,\vec{a}_n\}$ be a set of vectors in an inner product space
$V$. Than the {\it Gramian G} of this set is given by: $G_{ij}=(\vec{a}_i,\vec{a}_j)$.
The set of vectors is independent if and only if $\det(G)=0$.
A set is {\it orthonormal} if $(\vec{a}_i,\vec{a}_j)=\delta_{ij}$.
If $\vec{e}_1,\vec{e}_2,...$ form an orthonormal row in an infinite dimensional
vector space Bessel's inequality holds:
\[
\|\vec{x}\|^2\geq\sum_{i=1}^\infty|(\vec{e}_i,\vec{x})|^2
\]
The equal sign holds if and only if
$\lim\limits_{n\rightarrow\infty}\|\vec{x}_n-\vec{x}\|=0$.
The inner product space $\ell^2$ is defined in $\mathbb{C}^\infty$ by:
\[
\ell^2=\left\{\vec{a}=(a_1,a_2,...)~|~\sum_{n=1}^\infty|a_n|^2<\infty\right\}
\]
A space is called a {\it Hilbert space} if it is $\ell^2$ and if also holds:
$\lim\limits_{n\rightarrow\infty}|a_{n+1}-a_n|=0$.
Than there exists a Laplace transform for $f$.
The Laplace transformation is a generalisation of the Fourier transformation.
The Laplace transform of a function $f(t)$ is, with $s\in\mathbb{C}$ and $t\geq0$:
\[
F(s)=\int\limits_0^\infty f(t){\rm e}^{-st}dt
\]
The Laplace transform of the derivative of a function is given by:
\[
{\cal L}\left(f^{(n)}(t)\right)=-f^{(n-1)}(0)-sf^{(n-2)}(0)-...-s^{n-1}f(0)+s^nF(s)
\]
The operator $\cal L$ has the following properties:
If $s\in I\hspace{-1mm}R$ than holds $\Re(\lambda f)={\cal L}(\Re(f))$ and
$\Im(\lambda f)={\cal L}(\Im(f))$.
For some often occurring functions holds:
5.3.2 Matrix equations
We start with the equation
\[
A\cdot\vec{x}=\vec{b}
\]
and $\vec{b}\neq\vec{0}$. If $\det(A)=0$ the only solution is $\vec{0}$. If
$\det(A)\neq0$ there exists exactly one solution $\neq\vec{0}$.
5.4 Linear transformations
A transformation $A$ is linear if:
$A(\lambda\vec{x}+\beta\vec{y})=\lambda A\vec{x}+\beta A\vec{y}$.
Transformation type Equation
Projection on the line $<\vec{a}>$ $P(\vec{x})=(\vec{a},\vec{x})\vec{a}/(\vec{a},\vec{a})$
Projection on the plane $(\vec{a},\vec{x})=0$ $Q(\vec{x})=\vec{x}-P(\vec{x})$
Mirror image in the line $<\vec{a}>$ $S(\vec{x})=2P(\vec{x})-\vec{x}$
Mirror image in the plane $(\vec{a},\vec{x})=0$ $T(\vec{x})=2Q(\vec{x})-\vec{x}=\vec{x}-2P(\vec{x})$
5.5 Plane and line
The equation of a line that contains the points $\vec{a}$ and $\vec{b}$ is:
\[
\vec{x}=\vec{a}+\lambda(\vec{b}-\vec{a})=\vec{a}+\lambda\vec{r}
\]
The equation of a plane is:
\[
\vec{x}=\vec{a}+\lambda(\vec{b}-\vec{a})+\mu(\vec{c}-\vec{a})=\vec{a}+\lambda\vec{r}_1+\mu\vec{r}_2
\]
When this is a plane in $I\hspace{-1mm}R^3$, the normal vector to this plane is given
by:
\[
\vec{n}_V=\frac{\vec{r}_1\times\vec{r}_2}{|\vec{r}_1\times\vec{r}_2|}
\]
A line can also be described by the points for which the line equation
$\ell$: $(\vec{a},\vec{x})+b=0$ holds, and for a plane V: $(\vec{a},\vec{x})+k=0$.
The normal vector to V is than: $\vec{a}/|\vec{a}|$.
5.6 Coordinate transformations
The linear transformation $A$ from $I\hspace{-1mm}K^n\rightarrow I\hspace{-1mm}K^m$ is given by
($I\hspace{-1mm}K=I\hspace{-1mm}R$ or $\mathbb{C}$):
\[
\vec{y}=A^{m\times n}\vec{x}
\]
where a column of $A$ is the image of a base vector in the original.
5.7 Eigen values
The eigenvalue equation
\[
A\vec{x}=\lambda\vec{x}
\]
with eigenvalues $\lambda$ can be solved with
$(A-\lambda I\hspace{-1mm}I)=\vec{0}\Rightarrow\det(A-\lambda I\hspace{-1mm}I)=0$. The eigenvalues
follow from this characteristic equation. The following is true:
$\det(A)=\prod\limits_i\lambda_i$ and
${\rm Tr}(A)=\sum\limits_ia_{ii}=\sum\limits_i\lambda_i$.
5.8 Transformation types
5.8.1 Isometric transformations
A transformation is isometric when: $\|A\vec{x}\|=\|\vec{x}\|$. This
implies that the eigen values of an isometric transformation are given by
$\lambda=\exp(i\varphi)\Rightarrow|\lambda|=1$. Than also holds:
$(A\vec{x},A\vec{y})=(\vec{x},\vec{y})$.
5.8.2 Orthogonal transformations
A transformation $A$ is {\it orthogonal} if $A$ is isometric and
the inverse $A^\leftarrow$ exists. For an orthogonal transformation $O$ holds
$O^TO=I\hspace{-1mm}I$, so: $O^T=O^{-1}$. If $A$ and $B$ are orthogonal, than $AB$ and
$A^{-1}$ are also orthogonal.
5.8.3 Unitary transformations
Let $V$ be a complex space on which an inner product is defined. Than a linear
transformation $U$ is {\it unitary} if $U$ is isometric {\it and} its inverse
transformation $A^\leftarrow$ exists. A $n\times n$ matrix is unitary if
$U^HU=I\hspace{-1mm}I$. It has determinant $|\det(U)|=1$. Each isometric transformation
in a finite-dimensional complex vector space is unitary.
5.8.4 Symmetric transformations
A transformation $A$ on $I\hspace{-1mm}R^n$ is symmetric if
$(A\vec{x},\vec{y})=(\vec{x},A\vec{y})$. A matrix $A\in I\hspace{-1mm}M^{n\times n}$
is symmetric if $A=A^T$. A linear operator is only symmetric if its matrix
w.r.t. an arbitrary basis is symmetric. All eigenvalues of a symmetric
transformation belong to $I\hspace{-1mm}R$. The different eigenvectors are mutually
perpendicular. If $A$ is symmetric, than $A^T=A=A^H$ on an orthogonal basis.
5.8.5 Hermitian transformations
A transformation $H:V\rightarrow V$ with $V=\mathbb{C}^n$ is Hermitian if
$(H\vec{x},\vec{y})=(\vec{x},H\vec{y})$. The Hermitian conjugated
transformation $A^H$ of $A$ is: $[a_{ij}]^H=[a_{ji}^*]$. An alternative
notation is: $A^H=A^\dagger$. The inner product of two vectors $\vec{x}$ and
$\vec{y}$ can now be written in the form: $(\vec{x},\vec{y})=\vec{x}^H\vec{y}$.
5.8.6 Normal transformations
For each linear transformation $A$ in a complex vector space $V$ there exists
exactly one linear transformation $B$ so that $(A\vec{x},\vec{y})=(\vec{x},B\vec{y})$.
This $B$ is called the adjungated transformation of $A$. Notation:
$B=A^*$. The following holds: $(CD)^*=D^*C^*$. $A^*=A^{-1}$ if $A$ is unitary
and $A^*=A$ if $A$ is Hermitian.
and complex numbers $\alpha_1,...,\alpha_p$ so that
$A=\alpha_1P_1+...+\alpha_pP_p$.
5.8.7 Complete systems of commuting Hermitian transformations
Consider $m$ Hermitian linear transformations $A_i$ in a $n$ dimensional
complex inner product space $V$. Assume they mutually commute.
5.9 Homogeneous coordinates
Homogeneous coordinates are used if one wants to combine both rotations and
translations in one} matrix transformation. An extra coordinate is
introduced to describe the non-linearities. Homogeneous coordinates are derived
from cartesian coordinates as follows:
\[
\left(\begin{array}{c}x\\ y\\ z\end{array}\right)_{\rm cart}=
\left(\begin{array}{c}wx\\ wy\\ wz\\ w\end{array}\right)_{\rm hom}=
\left(\begin{array}{c}X\\ Y\\ Z\\ w\end{array}\right)_{\rm hom}
\]
so $x=X/w$, $y=Y/w$ and $z=Z/w$. Transformations in homogeneous coordinates
are described by the following matrices:
5.10 Inner product spaces
A complex inner product on a complex vector space is defined as follows:
5.11 The Laplace transformation
The class LT exists of functions for which holds:
$f(t)=$ $F(s)={\cal L}(f(t))=$
$\displaystyle\frac{t^n}{n!}{\rm e}^{at}$ $(s-a)^{-n-1}$
${\rm e}^{at}\cos(\omega t)$ $\displaystyle\frac{s-a}{(s-a)^2+\omega^2}$
${\rm e}^{at}\sin(\omega t)$ $\displaystyle\frac{\omega}{(s-a)^2+\omega^2}$
$\delta(t-a)$ $\exp(-as)$
If ${\cal L}(f)=F_1\cdot F_2$, than is $f(t)=f_1*f_2$.
Assume that $\lambda=\alpha+i\beta$ is an eigenvalue with eigenvector $\vec{v}$, than $\lambda^*$ is also an eigenvalue for eigenvector $\vec{v}^*$. Decompose $\vec{v}=\vec{u}+i\vec{w}$, than the real solutions are \[ c_1[\vec{u}\cos(\beta t)-\vec{w}\sin(\beta t)]{\rm e}^{\alpha t}+c_2[\vec{v}\cos(\beta t)+\vec{u}\sin(\beta t)]{\rm e}^{\alpha t} \]
There are two solution strategies for the equation $\ddot{\vec{x}}=A\vec{x}$:
Starting with the equation \[ ax^2+2bxy+cy^2+dx+ey+f=0 \] we have $|A|=ac-b^2$. An ellipse has $|A|>0$, a parabola $|A|=0$ and a hyperbole $|A|<0$. In polar coordinates this can be written as: \[ r=\frac{ep}{1-e\cos(\theta)} \] An ellipse has $e<1$, a parabola $e=1$ and a hyperbola $e>1$.
Rank 2: \[ p\frac{x^2}{a^2}+q\frac{y^2}{b^2}+r\frac{z}{c^2}=d \]
Rank 1: \[ py^2+qx=d \]