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Browse Classroom Capsules and Notes

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Displaying 41 - 50 of 85

There are three somewhat distinct topics in this article: the condition for linear functions to commute, a linear function as a transformation of the number line, and linear difference...

This article provides a non-group theory approach to finding the number of two by two matrices over \( Z/pZ\) that have both eigenvalues in the same field.  The strategy is to use the...

The fact that the limit of the \(n\)-th power of a \(2\times 2\) matrix \(A\) tends to \(0\) if  \( \det A < 1\) and \( \mid 1 + \det(A) \mid > \mid\) tr\( (A) \mid \) is used...

The author shows that a finite group \(G\) can be represented as a group of invertible \( 2 \times 2\) integer matrices if and only if \(G\) is isomorphic to a subgroup of the dihedral groups...

Consider a tournament with \(n\) players where each player plays every other player once, and ties are not allowed.  An \( n \times n\) tournament matrix \(A\) is constructed where...

If noise in data transmission produces a not quite orthogonal matrix that is known to be orthogonal, how does one find the "nearest" orthogonal matrix?  This capsule recasts the...

The article supplies a short, elementary proof that for integers \(a_1 < a_2 < \cdots < a_n \), the expression \( \prod_{n \geq i > j \geq 1} \frac{a_i - a_j}{i-j} \) is an integer...

Given symmetric unit vectors \(u_i\), conditions on real numbers \(x_i\) are considered to be able to conclude that \(\sum x_i u_i \Rightarrow x_i = 0\) for all \(i\). Different kinds of...

The author shows that the volume of an ellipsoid can be determined by three parallel slices.

Given any 3 by 3 rotation matrix \(A\) (i.e. orthogonal with determinant \(1\) and an arbitrary vector \(x\), the vector \( Ax +A^{T}x+[1−\)tr\((A)]x\) lies on the axis of rotation. The...

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