Class 3: Inverses and Determinants*
Inverses
Given a matrix A the generalized right inverse of A is defined by a matrix B such that
AB = I
(3.1)
( )
where I is the n n identity matrix ij . If A is n p then B must be p n . Let the n
columns of the matrix B
Class 1: Vector Spaces*
Vectors arise naturally in the physical sciences as quantities that require the specification
of both magnitude (length) and direction. Typical examples include force and velocity.
For a vector v, the norm (length or magnitude) is
Class 6: Exponential matrix function and non-diagonalizable systems*
When the matrix is not diagonalizable it is convenient to introduce the exponential matrix
function. This is defined formally by
A2
An
.
e =I+A+
+ . =
2!
n =0 n !
A
(6.1)
It follows tha
Class 2: Matrices*
Linear independence and spanning sets
In discussing a minimal spanning set or basis, it is necessary to eliminate all linearly
dependent vectors from the set, i.e., if a set of vectors a j are linearly independent then the
only solution
Class 5(a): Quadratic forms*
Quadratic forms
If A is a square matrix, a scalar quantity of the type
q ( x1 ,., xn ) = xT Ax = aij xi x j
i
(5.1)
j
is called a quadratic form. They arise in various physical systems and are often associated
with some energy
Class 4: Eigenvalues and eigenvectors*
In the physical sciences it is often necessary to solve systems of linear differential
equations. A simple first order equation is
dx
= x
dt
(4.1)
for which the general solution is
x = ket .
(4.2)
dx1
= a11 x1 + a12