Search results

  1. M

    Why are positive definite matrices useful?

    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...
  2. M

    How to show I_n + A is invertible

    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 the expansion is only defined for non-negative integers?
  3. M

    How to generalize determinant and cross product

    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)
  4. M

    What _is_ an indefinite integral?

    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...
  5. M

    Elegant solution to this vector equation?

    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?
  6. M

    Elegant solution to this vector equation?

    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...
  7. M

    Meaning of A^T A and A A^T

    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?
  8. M

    Derivative of vectors?

    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}, ...
  9. M

    Extracting yaw, pitch, roll from transformation matrix

    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:
  10. M

    Projection onto a subspace

    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).
  11. M

    Trouble with function definition

    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"?
  12. M

    Trouble with function definition

    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...
  13. M

    Calculate coordinate in 3D triangle.

    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
  14. M

    Convert Two Vectors To each Other !

    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)...
  15. M

    Inverted shape?

    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.
  16. M

    Help with Rolle's theorem

    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...
  17. M

    Tangent to reparameterized curve

    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)?
  18. M

    Tangent to reparameterized curve

    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...
  19. M

    Solutions of Ax = b (for singular A)

    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...