Inner product of two matrices

Let M(m,n)[R] be the vector space of m by n matrices with elements in R and scalar field R.

Let A, B, C elements in M(m,n)[R] and x,y elements in R. Define the map

f: V x V -> R by (A,B) |-> Tr(B^T * A).

Since
- (xA+yB,C) = x(A,C) + y(B,C)
- (A,B) = (B,A)
- (A,A) >= 0
(proofs are trivial)

f is an inner product.

The psychology of Charlie Eppes.

He is often shown "stimming". "Stimming" is basically self-stimulation, a repetitive sound or motion made by people who have difficulty integrating all the sensory input around them as a buffer between themselves and the sensory stimulus.

Charlie has sensory integration issues. This means he has difficulty integrating sensory input around him into a cohesive whole, something that most of us do automatically but is sometimes an exhausting task for individuals in the autism spectrum. Charlie wears enormous headphones with no music playing while figuring out a math problem.

Charlie fails to see the big picture at almost every turn and has extreme difficulty anticipating spontaneous behavior. Charlie is most comfortable with inductive reasoning, and despite his high intelligence levels, finds deductive reasoning difficult at best.

Charlie struggles to interpret social cues from others, when he picks up on them at all. Charlie rarely intuits when he annoys everyone around him, either with his single-minded preoccupation with mathematics or his inability to accurately read the nuances of social situations. He is very literal minded, often rambles past the point where anyone is listening, and in fact, sometimes DOES NOT NOTICE everyone has stopped listening.

Charlie fails to connect his concrete theorizing to abstract human reality, and in fact, doing so is deeply disturbing to him.

Charlie displays an extremely limiting and persistent social naivete that also can translate as extreme social immaturity. He is 27, doesn't date, doesn't notice that his very pretty friend is probably interested in him.

There is the compulsive and completely preoccupying nature of Charlie's obsession with discrete mathematics.

( This entry is a summary of this article on "tv makes you stupid". )

Projection Matrix

A matrix P is a projection matrix IFF P=P*P.

Henry Pitcher

92-year-old to graduate from UH with a bachelor degree in math Crossing the stage to graduate from the University of Houston, summer 2007 semester culminated a quest that Henry Pitcher, 92, began 75 years ago. Full Story

Collatz' original drawing

Math and art

Although I seriously doubt if mathematics and art are even close I added http://math-art.net/ to the 'Cool Sites' list. I do agree that math pics can be beautiful. "... Where Mathematics and Art blends into a zen-like state of peace ...".

Venn diagram

Great Britain, The United Kingdom, or is it England: confused? Charlie Eppes ( Numb3rs ) would probably give a mini-lecture on Set Theory before drawing a Venn diagram to explain the difference. We all use math every day.

Collatz conjecture

The 3x + 1 problem or Collatz conjecture is simple in its formulation but until now no proof is known that it is in fact true for any n. A conjecture is generally believed true but a formal proof is still to be found.

The conjecture says that if you repeatedly apply the following rule to a number the end result will always be one (1):
- if even then divide by 2;
- if odd then multiply by 3 and add 1.

Example:
9
28
14
7
22
11
34
17
52
26
13
40
20
10
5
16
8
4
2
1

Take any number and try it.

What about 27?

Quadratic residu

A number q is called a quadratic residue modulo n if there exists an integer x such that x^2 = q mod n.

For example

a  a^2  a^2 mod 7
1  1    1
2  4    4
3  9    2
4  16   2
5  25   4
6  36   1
7  49   0
8  64   1
9  81   4
etc.

The quadratic residues mod 7 are 1,2 and 4. The set {1,2,4} is a group under multiplication mod 7:

* mod 7   1   2   4
      1   1   2   4
      2   2   4   1
      4   4   1   2