I have been learning about matrices, and am interested in their use as maps. I have seen that there is in fact a matrix for transforming to spherical and cylindrical polar coordinates (Wikipedia), whose determinant gives the scaling factor of the volume element (which is slightly confusing to me since I thought this was the role of the Jacobian, and the transformation matrix doesn't look like the Jacobian. Anyway...)

I was wondering if there are maps that matricees cannot represnet. I am sure there are in abstract mathematics, but I mean mappings that would be useful in physical problems such as that from cartesian coordinates to spherical or cylindrical polars, or some other curvilinear coordinate. Has this to do with the linearity of matrix transformations? From what I understand, matrices all represent linear maps because they follow the two rules of linear transformations.

EDIT: I have now seen this post in which it is said that non-linear transformations cannot be respresented by matrices, except by going to an n+1 dimensional matrix. So I suppose that answers part of my question. The things that are still not clear to me are:

  • Is it the case that EVERY matrix represnets a linear transformation?
  • and, can EVERY linear transformation be respresented by a matrix?
  • What restrictions are there as to which non-linear transformations can be represented by matrices (or indeed, which non-linear transformations can be represented by matrices if not all can).
  • 1
    $\begingroup$ There are some matrices shown in the Wikipedia article you link to -- but they are (and are explicitly stated to be) Jacobians, not matrices that represent the entire transformation. $\endgroup$ – Henning Makholm May 7 '17 at 10:59

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Basis}{\mathbf{e}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}\newcommand{\Proj}{\mathbf{P}}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}$For definiteness, let's work in the Cartesian vector space $\Reals^{n}$.

Theorem: A mapping $T:\Reals^{n} \to \Reals^{n}$ is linear if and only if there exists an $n \times n$ real matrix $A$ such that $T(x) = Ax$ for all $x$ in $\Reals^{n}$.

Proof (Sketch): If $A$ is an $n \times n$ real matrix, the mapping $T:\Reals^{n} \to \Reals^{n}$ defined by $T(x) = Ax$ is linear by properties of matrix multiplication. For all $x$, $y$ in $\Reals^{n}$ and all real $c$, $$ T(cx + y) = A(cx + y) = A(cx) + Ay = c(Ax) + Ay = cT(x) + T(y). $$ Conversely, if $T$ is linear, and if $(\Basis_{j})_{j=1}^{n}$ denotes the standard basis, the matrix $A$ whose $j$th column is the coordinate vector $T(\Basis_{j})$ satisfies $$ T(x) = \sum_{j=1}^{n} x_{j} T(\Basis_{j}) = Ax. $$

(There is an obvious generalization to linear transformations $T:\Reals^{n} \to \Reals^{m}$ and $m \times n$ real matrices.)

If $H \subset \Reals^{n+1}$ is an affine subspace, and if $T:\Reals^{n+1} \to \Reals^{n+1}$ is a linear transformation such that $T(H) \subseteq H$, then the restriction of $T$ to $H$ is a (possibly non-linear) transformation that can (in a sense) be represented by matrix multiplication.

The standard example is the set of affine transformations, $T(x) = Ax + b$, acting on the affine hyperplane $x_{n+1} = 1$, which may be effected by multiplication with a block $(n + 1) \times (n + 1)$ matrix: $$ \left[\begin{array}{cc} A & b \\ 0 & 1 \\ \end{array}\right] \left[\begin{array}{cc} x \\ 1 \\ \end{array}\right] = \left[\begin{array}{cc} Ax + b \\ 1 \\ \end{array}\right]. $$

If we relax the hypothesis that the transformation is defined everywhere in $H$, there is also the set of projective transformations (of the real projective space $\Reals\Proj^{n}$), whose component functions (in each affine chart) may be expressed as ratios of polynomials of degree $1$.

For instance, if $H$ is the hyperplane $x_{n+1} = 1$, and if we define $\Pi:\Reals^{n+1}\setminus \Reals^{n} \to H$ by $$ \Pi(x_{1}, \dots, x_{n}, x_{n+1}) = \frac{1}{x_{n+1}}(x_{1}, \dots, x_{n}, x_{n+1}) = \left(\frac{x_{1}}{x_{n+1}}, \dots, \frac{x_{n}}{x_{n+1}}, 1\right), $$ then for every linear transformation $T:\Reals^{n+1} \to \Reals^{n+1}$, the composition $$ \Pi \circ T:H \to H $$ is a (partial) transformation, defined everywhere that $T(x) \cdot \Basis_{n+1} \neq 0$.

Affine and projective transformations preserve lines. In the sense you're asking, a transformation represented by a matrix preserves lines.


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