Eigen values and Eigen vectors

In linear algebra the concept of Eigen values and Eigen Vectors relate to how we as mathematicians define a vector space. In this vector space, we define scalars which are just numbers and vectors which have an associated direction and magnitude. Often times we use matrices to represent how the space is set up. If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction, then the vector is called an eigenvector of that matrix. A corresponding scalar is associated with this vector and is called the eigen value. This is a rather abstract concept but it helps us map and quantify a space with many dimensions. In simple 2-d we can say eigen values of 1 to each of two eigen vectors (1,0) and (0, 1). Computers use these to shift and map images as well as alter them with great precision.

Birthday Matches

Let us suppose you have a class with about 35 students. Begin by asking

the class what they think the chances (or probability) are of two classmates

having the same birth date (month and day only) in their class of about

30+ students. Students usually begin to think about the likelihood of two

people having the same date out of a selection of 365 days (assuming no

leap year). Perhaps 2 out of 365?

The class will probably be surprised to learn that for a group of 35 the

probability that two members will have the same birth date is greater than

8 out of 10, or 80%

The probability that another student does not match the first student is:

364/365

The probability that a third student does not match the first and second

students is

:

363/365

The probability of all 35 students not having the same birth date is the

product of these probabilities:

(365/365)(364/365)(363/365)***(365-35)/(365)

Since the probability (q that two students in the group have the same

birth date and the probability (p) that two students in the group do not

have the same birth date is a certainty, the sum of those probabilities must

be 1. Thus, p + q = 1.

≈ 0.814383238874715

In other words, the probability that there will be a birth date match in a

randomly selected group of 35 people is somewhat greater than 80%. This

is quite unexpected when one considers there were 365 dates from which

to choose. Students may want to investigate the nature of the probability

function. Here are a few values to serve as a guide:

Number of people Probability of a birth

in group date match

10 0.1169481777110776

15 0.2529013197636863

20 0.4114383835805799

25 0.5686997039694639

30 0.7063162427192686

35 0.8143832388747152

40 0.891231809817949

45 0.9409758994657749

50 0.9703735795779884

55 0.9862622888164461

60 0.994122660865348

65 0.9976831073124921

70 0.9991595759651571

Students should notice how quickly almost-certainty is reached. With

about 60 students in a room, the table indicates that it is almost certain

(0.99) that two students will have the same birth date.

Were one to do this with the death dates of the first 35 presidents, one

would notice that two died on March 8 (Millard Fillmore in 1874 and

William H. Taft in 1930) and three presidents died on July 4 (John Adams

and Thomas Jefferson in 1826 and James Monroe in 1831).

Above all, this astonishing demonstration should serve as an eye-opener

about the inadvisability of relying entirely on intuition

Linear Algebra

Today, June 1 I have a Linear Algebra final. In the spirit of the subject and trying to simultaneously study I'll talk about some subjects in the field.

I. Matrices

-Definition: A matrix is a rectangular array of entries wherein the entry can be a number or a variable.

-Uses: Matrices can be used to solve systems of equations.

-Example of a matrix:

( 1 3 4 5) -> a 1x4 matrix

-Example of solving a system:

This represents the system: x+9y=13, 20x+55y=4y using the properties of matrices, we can reduce the matrix to get x by itself and simultaneously get y by itself. x=-5.432 y=2.048

II. Determinants

-Definition: a determinant is a unique number assigned to only a square matrix. note a square matrix is a matrix with equal number of rows as columns

-Uses: Easier ways of solving systems, Finding the inverses of a matrix

-Example: from the previous matrix delete the last column to get a 2x2 matrix. the determinant of that matrix would be: -125

III. Vector Spaces

-Definition: a vector space is a set on which vector addition and scalar multiplication are defined, and certain axioms hold.

-example: a common vector space is the set of all ordered pairs of real numbers with vector operations is a vector space

IV. Inner Product Spaces

-definition:an inner product of of a vector space associates a real number with a pair of vectors

-example: a simple example that is already defined is a the dot product, which is just the components of a vector multiplies then all added up.

V. Linear Transformations

-Definition: a Linear transformation preserves the operations on a set but alter the starting position of the basis vectors.

-uses: you got me...

VI. Eigen values/vectors

-Eigen values are a large topic with a wide range of uses and nuances. to try to summarize it here would still be a large effort with more text you care for. I'll make a separate post about that later