Some of the things I've learned every day since Oct 10, 2016
Monthly Archives: October 2016
October 31, 2016Posted by on
Because the computation of matrix determinants by cofactor expansion (Laplace expansion) is computationally very expensive — for example, computing the determinant of an matrix by this method requires over multiplications — other methods are usually used.
For instance, the simple method of using Gaussian elimination to reduce the matrix to its upper-triangular form, then taking the product of the diagonal to obtain the determinant, is just in multiplications. Other, even less expensive algorithms exist as well, such as the Bareiss algorithm.
October 30, 2016Posted by on
If is a linear operator on a finite-dimensional vector space , and the characteristic polynomial of splits, then for a given eigenvalue of with multiplicity , the generalized eigenspace corresponding to is
October 29, 2016Posted by on
The distribution of a discrete random variable is considered a Poisson Distribution if, given an event an given interval (which can be thought of a number of chances is given to occur), the probability of occurring exactly times on that size interval is given by
where is the known average of occurrences given an identical interval.
[Restrictions: events must occur independently, and no more than one instance of an event can happen at a single point in the interval.]
A couple nice properties of Poisson-distributed random variables are that the expected value and variance are both equal to .
October 28, 2016Posted by on
Kleene’s Theorem states that a set is regular if and only if it is recognized by a finite-state automaton.
October 27, 2016Posted by on
The expectation of a discrete random variable , denoted , is linear. That is, for random variables and a constant ,
October 26, 2016Posted by on
In the Lisp language, all code is written in the form of S-Expressions, which are essentially expressions in the form of parenthesized lists. For example, the expression of a function f being called on arguments a, b would be (f a b). The fact that code in Lisp is expressed in the form of the list data structure allows code to be manipulated as data, facilitating easy metaprogramming.
October 25, 2016Posted by on
A context-free grammar is a formal grammar in which the rewrite rules (or ‘productions’) of the grammar are all from single variables. That is, as implied in the name, the rewrite rules of a given variable in a context-free grammar are dependent only on that variable and not on any other ‘context’.
This is a valid subset of rewrite rules in a context-free grammar
while this is not
The violation of freedom-from-context in the latter example is due to the last rule, where the thing being rewritten is not a single variable.
October 24, 2016Posted by on
The Kolmogorov Complexity of something is, naively speaking, the minimum length of a program which produces that thing as output. Even more naively, it can be thought of as the amount of information necessary in a ‘description’ of an object.
October 23, 2016Posted by on
The Schläfli symbol of the dual of a polytope with dimension will simply be the reverse of that polytope’s symbol. Thus, such a polytope is self-dual if and only if its Schläfli symbol is palindromic.
October 22, 2016Posted by on
The Cayley-Hamilton Theorem states that a linear operator satisfies (is annihilated by) its own characteristic polynomial.
That is, if is a linear operator and
is its characteristic polynomial, then
where is the zero transformation.