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- Multiplying Sparse Matrices [12/01/2000]
Is there a reasonably fast way to multiply two matrices that are very
big, but sparse (they have many zeros)?
- Natural Numbers Coprime to 6 [02/16/1998]
Let N(x) denote the number of natural numbers less than x which are
coprime to 6. Show that lim as x goes to infinity of [N(x)/x] = 1/3.
- Nearest Point on a Great Circle [05/27/2002]
Given points A, B, and C on the surface of a unit sphere, find the
point P on the great circle defined by A and B that is nearest to C.
- One-to-one Proof [06/25/1998]
Let T(x) = Ax, where A is a 2 X 2 matrix. Show that T is one-to-one if
and only if the determinant of A is not zero.
- Orthogonal Distance Regression Line [07/05/2005]
How can I calculate the standard deviation of the error in the
independent and dependent variables when I want to fit a straight line
with both variables subject to error?
- Orthogonal Distance Regression Planes [07/30/2003]
I have a set of data points that I have collected from an experiment.
I want to fit a 3D plane (best-fit) into these points (the points are
in the form (x1,y1,z1), (x2,y2,z2),...) in order to evaluate my
- Orthogonal Matrices [01/07/1998]
Prove that if A is an orthogonal matrix, the determinant of A is either 1
- Pane Rotation - Computer Graphics [11/14/1996]
How do I find a formula for rotating a pane in space?
- Parity of Permutations [7/8/1996]
I can't remember how to prove that permutations are either odd or even.
- A Perfectly Reflecting Sphere [09/09/2000]
How can I find the coordinates of each intersection of a laser beam fired
from one of the "poles" of a perfect sphere into a perfectly mirrored
interior of the sphere?
- Point of Intersection between Two Lines [07/08/2003]
Given lines with coordinates (x1,y1) and (x2,y2), and (x3,y3) and
(x4,y4), find whether (and where) these lines intersect between
- Point within a Triangle [05/29/2003]
I have the coordinates of the three corners of a equilateral triangle
ABC. How can I decide whether an arbitrary point (X,Y) lies in the
plane of the triangle?
- Population Dynamics and the Leslie Matrix [11/04/2000]
What are the Leslie matrix and the age histogram vector for a population
- Positive Definite Matrices [08/12/2008]
How can I tell if a matrix is positive definite, and if it's not, how
can I make it be so?
- Powers of Matrices [6/27/1996]
Would you send me some information/links about matrices and exponentials?
- Principal Axis Theorem [12/10/1997]
Could you give me applications of the principal axis theorem?
- Product of Upper Triangular Matrices [09/23/2003]
Show that the product of two upper triangular matrices is an upper
- Proof of Circle Theorem by Vectors [05/03/2001]
Prove, using the vector scalar product, that the angle in a semicircle is
always 90 degrees (the hypotenuse being the diameter, and the sides
meeting on the perimeter).
- Proof of Interesting Geometric Vector Theorem [09/17/2005]
O is the center of the circumscribing circle of triangle ABC and H is
its orthocenter. Prove that vector OH is equal to the sum of the
vectors OA, OB and OC.
- Proof of Reflective Property of the Hyperbola [07/08/2004]
A discussion of how to prove that a ray of light aimed at one focus of
a hyperbola which hits one 'arm' will reflect off that arm and reach
the other focus if we imagine that the arms function as mirrors.
- Proofs on Idempotent Matrices [05/09/2000]
How can I determine all idempotent 2x2 matrices over R, and how can I
prove that in an integral domain, the only idempotents are 0 and 1?
- Properties of Determinants [10/23/2000]
What properties of determinants do we use to evaluate them, and how? How
do we put determinants in row-echelon form?
- Proving Concurrence Using Vectors [10/17/2005]
How do you prove that angle bisectors are concurrent using vectors? I
have proved this using coordinate geometry, but I do not know how to
find the point of intersection using vectors.
- Rotating a Plane about a Point in Space [07/03/2004]
Discussion of a method of rotating the three points of a right
triangle around a point to align with a given plane in a particular way.
- Rotating Vectors [12/14/1995]
If I were a vector and I wanted to turn to face another vector how would
I do it?
- Rotation Matrix about a Point Other Than the Origin [12/18/2006]
I know that in order to rotate a point around the origin you use the
rotation matrix with sine and cosine. I was wondering how you rotate a
point around a point other than the origin?
- Rotation Matrix Using Trig Functions [12/15/2005]
I know what the rotation matrix is for an angle of rotation of theta,
but I don't understand why it works.
- Rotations in Three Dimensions [11/11/1999]
How can I rotate something about an axis or about a line in 3D?
- Shortest Distance Between Two Vectors [04/07/1998]
How do you find the shortest distance between two 3-dimensional lines?
- Singular Value Decomposition [11/04/2005]
Can you explain what kinds of difficulties may arise while trying to
find a singular value decomposition? For example, can we find
eigenvectors for corresponding eigenvalues in every situation?
- Skew-Symmetry [02/17/1997]
Using skew-symmetry to prove statements about matrices.
- Solve Eigenvalue of Complex Matrix [7/17/1996]
Please tell me about the two methods to solve the eigenvalues of complex
matrices - the Hause-Gibbson method and the Jacobi method.
- Solving Linear Equations in Computer Programs [02/13/2003]
What is the best and simplest method in computer programs for solving
linear equations: Cramers method; Gaussian Elimination method; Inverse
- Some Algebra Problems [6/1/1996]
If z=(3-2i)^1/2 then find z^-...
- Special Unitary Groups In Physics [11/09/1998]
Could you explain the groups SU(2) and SU(3)? They are central to
descriptions of quantum chromodynamics.
- Square Root of Matrix A [06/29/2003]
How can we find the square root of matrix A... ?
- The Stationary Vector [11/13/2000]
In Markov processes, what does a stationary vector tell you and how do
you find it?
- Subspaces of R^3... [06/22/1998]
Find a basis and determine the dimension of the subspace...
- Symmetric Matrices [10/27/1999]
Find all values of a, b, and c for which a 3x3 matrix A is symmetric,
where the elements of A contain algebraic expressions in a, b, and c.
- Tensor [10/01/1997]
Is there a layman's definition for a tensor, with an example?