One reason is that if a matrix A is positive definite, the quadratic form
f(x) = \frac{1}{2} x A^T x + b^Tx + c
has a unique minimum (expressions like these crop up in a number of places). A positive definite matrix A can be visualized as a paraboloid (look at the graph of f) that is...
Hi all,
I tried to prove this for myself, but did not get anywhere :-(
I don't quite get this... how does the Binomial theorem help here? According to http://en.wikipedia.org/wiki/Binomial_theorem the expansion is only defined for non-negative integers?
If A is the matrix whose column vectors are X_1, ..., X_k, the "hypervolume" V of the parallelepiped spanned by the vectors is given by
V^2 = \det(A^TA)
Given an indefinite integral,
\int f(x) dx = F(x) + C,
I am having some problems in understanding what this indefinite integral "is". The RHS is clearly a function, but what is the LHS? Judging by the equals sign, it should also be a function, but seemingly it isn't because there's no...
This looks interesting... I tried to reverse engineer your solution, but didn't quite get there. How do you use the fact that the vectors you mention span \mathbb{R}^3?
Given are a plane E and a line l in general position. I need to find a plane that contains l and intersects E at a given angle \alpha. All of this happens in R^3.
The interesting part is to find the normal of the unknown plane, let us call this normal x. I came up with the following...
The matrices A^{T}A and AA^{T} come up in a variety of contexts. How should one think about them - is there a way to understand them intuitively, e.g. do they have a geometric interpretation?
For a symmetric matrix B (in your case, B = A^T A), the following is a scalar-valued function from R^n to R:
f(x) = x^T B x
The derivative you are looking for is defined as the vector of partial derivatives (aka gradient):
\frac{df}{dx} = \left(\frac{\partial f}{\partial x_1}, ...
This kind of conversion is rather ugly... a nice algorithm that handles all possible configurations of axes (including roll-pitch-yaw) very compactly is given here:
http://etclab.mie.utoronto.ca/people/david_dir/GEMS/GEMS.html
The problem is that A will be rectangular (non-square) if you are projecting onto a subspace, and thus its inverse does not exist (e.g. A is a column vector for projection onto a line).
Thanks for your answer, I think I understand it better now. The one thing that still bothers me is the \partial(ts). How is this to be interpreted? Is it just a placeholder that says "partial differentiation by the first argument"?
Given is the following function (nevermind what the function h is):
g(t, q) = \int_0^1 \frac{\partial h(ts, q)}{\partial(ts)} ds
This function is supposed to be defined for t = 0. However, I don't see how - the partial derivative in the integral then becomes \frac{\partial h(0...
You could set up the plane equation
ax + by +cz + d = 0
of the plane containing the triangle ABC, insert the known (x, z) coordinates of your point D and solve for y
Are these 2D or 3D vectors? In general, to find a rotation matrix R that maps a vector u to a vector v (assumed to be normalized) you need to create two orthogonal matrices U and V whose first columns are u and v, respectively. Make sure that the determinant is 1 in both cases (rather than -1)...
Calculate
\sum_{i=1}^n x_i y_{i+1} - x_{i+1} y_i
in which (x_i, y_i) are the coordinates of the vertices. This will give you twice the signed area of the polygon, i.e. if the sign is positive, the ordering is counter-clockwise, otherwise it is clockwise.
Given is a one-parameter family of planes, through
x \cdot n(u) + p(u) = 0
with normal vector n and base point p, both depending on the parameter u.
Two planes with parameters u_1 and u_2, with u_1 < u_2, intersect in a line (planes are assumed to be non-parallel). This line also lies...
Thank you very much, I think I can work it out now!
When you say my notation is bad, are you referring to my application of the chain rule (which I believe is flawed), or to the notation in which the problem was posed (which was taken from "Tensor analysis on manifolds", Bishop & Goldberg)?
Given is a curve \gamma from \mathbb{R} \rightarrow M for some manifold M. The tangent to \gamma at c is defined as
(\gamma_*c)g = \frac{dg \circ {\gamma}}{du}(c)
Now, the curve is to be reparameterized so that \tau = \gamma \circ f, with f defining the reparametrization. (f' > 0...
A system of linear equations, Ax = b (with A a square matrix), has a unique solution iff det(A) \ne 0. If b = 0, the system is homogeneous and can be solved using SVD (which gives the null space of A).
Now, how can the solution set be characterized for singular A and b \ne 0? If a single...