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Review of Matrix Operations

- Vector a sequence of elements (the order is

important) - e.g., x (2, 1) denotes a vector
- length sqrt(2211)
- orientation angle a
- x (x1, x2, , xn), an n dimensional vector
- a point in an n dimensional space
- column vector row vector

X (2, 1)

a

transpose

- norms of a vector (magnitude)
- vector operations

- Cross product
- defines another vector orthogonal to the

plan - formed by x and y.

- Matrix
- the element on the ith row and jth

column - a diagonal element (if m n)
- a weight in a weight matrix W
- each row or column is a vector
- jth column vector
- ith row vector

- a column vector of dimension m is a matrix of m x

1 - transpose
- jth column becomes jth row
- square matrix
- identity matrix

- symmetric matrix m n and
- matrix operations
- The result is a row vector, each element of which

is an inner product of and a column vector

- product of two matrices
- vector outer product

Linear Algebra

- Two vectors
- are said to be orthogonal to each other if
- A set of vectors of

dimension n are said to be linearly independent

of each other if there does not exist a set of

real numbers which are not all

zero such that - otherwise, these vectors are linearly dependent

and each one can be expressed as a linear

combination of the others

- Vector x ! 0 is an eigenvector of matrix A if

there exists a constant ? such that Ax ?x - ? is called a eigenvalue of A (wrt x)
- A matrix A may have more than one eigenvectors,

each with its own eigenvalues - Ex.

has 3 eigenvalues/eigenvectors

- Matrix B is called the inverse matrix of square

matrix A if AB I (I is the identity matrix) - Denote B as A-1
- Not every matrix has inverse (e.g., when one of

the row can be expressed as a linear combination

of other rows) - Every matrix A has a unique pseudo-inverse A,

which satisfies the following properties - AAA A AAA A AA (AA)T AA

(AA)T - Ex. A (2 1 -2), A (2/9 1/9 -2/9) T

- Calculus and Differential Equations
- , the derivative of , with respect to

time - System of differential equations
- solution
- difficult to solve unless are simple

- Multi-variable calculus
- partial derivative gives the direction and

speed of - change of y with respect to . Ex.

- the total derivative of
- gives the direction and speed of change of y,

with respect to t - Gradient of f
- Chain-rule z is a function of y, y is a

function of x, x is a function of t

- dynamic system
- change of may potentially affect other x
- all continue to change (the system evolves)
- reaches equilibrium when
- stability/attraction special equilibrium point
- (minimal energy state)
- pattern of at a stable state

often represents a solution of the problem

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