Let A and B be 2 × non-zero complex matrices. P a linear combination of in the form P=c_1 A+c_2 where c_1,c_2 are nonzero scalar numbers. An idempotent matrix is which, when multiplied by itself, yields itself. In this study, we established entries according to given such that also an matrix. addition, result was obtained determined singular

This paper introduces and analyzes an algorithm to compile a series of exclusive-or operations. The compiled series is quite efficient, almost always beating the so-called “Four Russians” approach, and uses no temporary storage beyond its outputs. The algorithm is reasonably fast and surprisingly simple.

Using the symmetric form of the Lovász Local Lemma, one can conclude that a k-uniform hypergraph H admits a proper 2-colouring if the maximum degree (denoted by ∆) of H is at most 2 8k independently of the size of the hypergraph. However, this argument does not give us an algorithm to find a proper 2-colouring of such hypergraphs. We call a hypergraph linear if no two hyperedges have more than ...

We start with the simple case of studying the relationship between a response variable Y and a predictor variable X I . Since we have only one predictor variable, we shall drop the subscript in X I and use X for simplicity. We discuss covariance and correlation coefficient as measures of the direction and strength of the linear relationship between the two variables. Simple linear regression mo...

For $n=2^k$, let $S$ be an $n x n$ matrix whose rows and columns are indexed by $GF(2)^k$ and, for $I,j \in GF(2)^k$, $S_{I,j}={I,j}$, the standard inner product. Size-depth trade-offs are investigated for computing $Sx$ with circuits using only linear operations. In particular, linear size circuits with depth bounded by the inverse of an Ackerman function are constructed, and it is shown that ...

The general approach to the foundations of mathematics is to study certain spaces, and then to study functions between these spaces. In this course we follow this paradigm. Up until now, we have been studying properties of vector spaces. Vector spaces have a linear structure, and so it is natural to deal with functions between vector spaces that preserve this linear structure. That is, we will ...