Application of Invertible Matrices: Coding
There are many ways to encrypt a message. And the use of coding has become
particularly significant in recent years (due to the explosion of the internet for
example). One way to encrypt or code a message uses ma
Diagonalization
When we introduced eigenvalues and eigenvectors, we wondered when a square
matrix is similarly equivalent to a diagonal matrix? In other words, given a square
matrix A, does a diagonal matrix D exist such that
invertible matrix P such that
Elementary Operations for Matrices
Elementary operations for matrices play a crucial role in finding the inverse or solving
linear systems. They may also be used for other calculations. On this page, we will
discuss these type of operations. Before we def
Complex numbers as Matrices
In this section, we use matrices to give a representation of complex numbers. Indeed,
consider the set
We will write
Clearly, the set
is not empty. For example, we have
In particular, we have
for any real numbers a, b, c, and d
Algebraic Properties of Matrix Operations
In this page, we give some general results about the three operations: addition,
multiplication, and multiplication with numbers, called scalar multiplication.
From now on, we will not write (mxn) but mxn.
Propert
Invertible Matrices
Invertible matrices are very important in many areas of science. For example,
decrypting a coded message uses invertible matrices (see the coding page). The
problem of finding the inverse of a matrix will be discussed in a different pa
Introduction and Basic Operations
Matrices, though they may appear weird objects at first, are a very important tool in expressing
and discussing problems which arise from real life cases.
Our first example deals with economics. Indeed, consider two famil
Special Matrices: Triangular, Symmetric,
Diagonal
We have seen that a matrix is a block of entries or two dimensional data. The size of
the matrix is given by the number of rows and the number of columns. If the two
numbers are the same, we called such ma
Multiplication of Matrices
Before we give the formal definition of how to multiply two matrices, we will discuss
an example from a real life situation. Consider a city with two kinds of population: the
inner city population and the suburb population. We a
Matrix Exponential
The matrix exponential plays an important role in solving system of linear differential
equations. On this page, we will define such an object and show its most important
properties. The natural way of defining the exponential of a matr
Markov Chains
In a previous page, we studied the movement between the city and suburbs. Indeed,
if I are S are the initial population of the inner city and the suburban area, and if we
assume that every year 40% of the inner city population moves to the s