How should one interpret statements of form $z^T M z$? From wikipedia:

In linear algebra, a symmetric $n \times n$ real matrix $M$ is said to be positive definite if the scalar $z^T M z$ is positive for every non-zero column vector $z$ of $n$ real numbers.

Question: Matrix multiplication is often interpreted as as the composition of linear transformations; that being case, how should one interpret statements of the form $z^T M z$? Are they something like
$$
v^{-1} T v
$$
where $T$ is a linear transformation $T : V \rightarrow V$ and $v \in V$?
 A: Anything expression involving a transpose requires some kind of "dot product" (more generally, a choice of isomorphism from $V$ to the dual space $V^*$).  What we're measuring, then, is the dot product between the "input vector" $z$ and the "transformed vector" $Mz$ to get $z^T(Mz)$.
In general, the map $z \mapsto z^TMz$ is called a "quadratic form".  So, hyper-surfaces of the form $z^TMz = 1$ are "hyperboloids" and "ellipsoids" in $n$-dimensional space.
A: Instead of quadratic forms you may also think it via bilinear forms. If $V$ is an $n$-dinensional real vector space with basis $\mathcal{B}$ any bilinear form $B:V\times V \to \mathbb{R}$ corresponds to a unique $n\times n$ matrix $M$ s.t. 
$$ B(\mathbf{x},\mathbf{y})= [\mathbf{x}]_\mathcal{B}^T M [\mathbf{y}]_\mathcal{B}$$
for every $\mathbf{x},\mathbf{y}\in V$. 
If we think a bilinear form as a generalization of the dot product then it's natural to consider if they are symmetric ($B(\mathbf{x},\mathbf{y})=B(\mathbf{y},\mathbf{x}) \forall \mathbf{x},\mathbf{y}\in V$) and whether $B(\mathbf{x},\mathbf{x}) > 0 \forall \mathbf{x}\not= \bar{0}$.
A: The form $z^T Mz$ is a quadratic form.  A linear form in $n$ variables $z_1,\ldots,z_n$ is a function of the form $f(z_1,\ldots,z_n)=c_1 z_1+\cdots+c_n z_n+d$ for some constant $d$ (in some field).  The function $f$ can be expressed using vector notation as $f(z)=c^T z + d$.  In many situations (such as physics, conic sections), the equations that arise are quadratic forms $Ax^2+By^2+Cxy$. This quadratic form can be expressed using matrices as $$Ax^2+By^2+Cxy =(x,y) \left( \begin{array}{cc} A & C/2 \\ C/2 & B \end{array} \right) \left( \begin{array}{c}x \\ y \end{array} \right).$$
Every real quadratic form can be expressed in the form $z^T Mz$ for some real symmetric matrix $M$ (the skew-symmetric part of $M$ can be removed because $z^TKz=0$ if $K$ is skew-symmetric). Though we see cross-terms like $xy$, these forms can be bought to diagonal form using certain change-of-basis transformations. We can think of completing the square as bringing the quadratic form to one which is represented by a diagonal matrix.
If the matrix $M$ in $z^TMz$ is real symmetric positive-definite, then the quadratic form $z^T M z = \sum_i \sum_j M_{ij} z_i z_j$ can be brought to a diagonal form $y^T D y = \sum_i d_{ii} y_i^2$ where the $d_{ii}$'s (eigenvalues of $M$) are strictly positive. This implies that whenever $z$ is nonzero, $y$ is also nonzero and so the value of the quadratic form is strictly positive. 
