A simple closed form for triangular matrix powers

Ela a Simple Closed Form for Triangular Matrix Powers∗

1. The algorithm. Huang [1] gives an algorithm for computing the powers of a triangular matrix where the diagonal elements are unique. However, in contrast to Huang’s algorithm, the method presented here has the unique advantage of producing the result in closed form, which shows explicitly how the behavior of any element of the matrix varies with varying powers of the matrix. Also, the closed ...

A simple closed form for triangular matrix powers

Closed-Form Expressions for the Matrix Exponential

We discuss a method to obtain closed-form expressions of f(A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley–Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to physicists. Here, we derive the results on which the method is based by using tools most commonly employed by physicists. W...

Rigidity of a simple extended lower triangular matrix

For the all-ones lower triangular matrices, the upper and lower bounds on rigidity are known to match [13]. In this short note, we apply these techniques to the allones extended lower triangular matrices, to obtain upper and lower bounds with a small gap between the two; we show that the rigidity is θ( 2 r ).

Closed form bounds for clock synchronization under simple uncertainty assumptions

Many applications in a distributed system rely on processors having synchronized local clocks. Many studies have been dedicated to algorithms and lower bounds for clock synchronization under various network assumptions. We consider the problem of synchronizing the clocks of processors in a failure-free distributed system when the hardware clocks do not drift but there is uncertainty in the mess...