MATH 223: Some results for 2 2 matrices.
One can preview a lot of the theory in this course by looking at the special case of 2 2
matrices. The proofs are relatively easy in this limited context and not all the complexity is seen.
In particular, the Gauss
MATH 223: Some results for 2 2 matrices.
Multiplicative Inverses
It would be nice to have a multiplicative inverse. That is given a matrix A, nd the inverse
matrix A1 so that AA1 = A1 A = I. Such an inverse can be shown to be unique, if it exists
(How?).
MATH 223 Systems of Dierential Equations including example with Complex Eigenvalues
First consider the system of DEs which we motivated in class using water passing through two
tanks ushing out salt.
1
y1 (t) = 10 y1 (t) +
1
y2 (t) = 10 y1 (t)
1
y (t)
40
MATH 223
Complex Numbers
When solving a quadratic (over R) you may nd
there are no roots but you notice that you get
expressions involving 1. Rather than interpret 1 as a number, we can proceed as follows.
We dene
C = cfw_a + bi : a, b R
and show that wit
Math 223
Symmetric and Hermitian Matrices.
An n n matrix Q is orthogonal if QT = Q1 . The columns of Q would form an orthonormal
basis for Rn . The rows would also form an orthonormal basis for Rn .
A matrix A is symmetric if AT = A.
Theorem Let A be a sy
MATH 223: Extremal Set Theory
I would like to demonstrate one proof of a result of Sauer, Perles and Shelah, Vapnik and
Chervonenkis from 1971,1972. The proof is due to Smolensky and is from 1997. There are a variety
of proofs, basic induction works ne.
T