7/26/2019 UMEP Sample
1/2
UMEP Mathematics Exam Revision Book
Copyright Victor Lin 2014 Linear Equations and Matrices 32
Topic 5: Linear Equations and Matrices
Linear equations and linear systems
A linear equation in n variables,1 2
, , ,n
x x x , is an equation of the form:
1 1 2 2 n na x a x a x b
where1 2
, , ,n
a a a are constants (not all zero) and b is also a constant.
Asystem of linear equations or a linear system is a finite
collection of linear equations. The
following is a system of m linear equations with n
variables:
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
n n
n n
m m mn n m
a x a x a x b
a x a x a x b
a x a x a x b
The system can be written as an augmented matrix:
11 12 1 1
21 22 2 2
1 2
n
n
m m mn m
a a a b
a a a b
a a a b
The number of rows in an augmented matrix is equal to the number
of equations. The number ofcolumns, discounting the last, is equal
to the number of variables. The last column contains theconstant
term from each equation.
Elementary row operations
The basic method for solving a system of linear equations is to
replace the given system by anew system that has the same solution
set but is easier to solve. This new system is generallyobtained in
a series of steps by applying the following three types of
operations to eliminateunknowns systematically:
1. Multiply an equation by a non-zero constant2.
Interchange two equations3. Add a multiple of one equation to
another equation.
Since the rows of an augmented matrix correspond to the
equations in the associated system,these three operations
correspond to the following operations on the rows of the
augmentedmatrix:
1. Multiply a row by a non-zero constant2. Interchange two
rows3. Add a multiple of one row to another row.
These three operations are known as the elementary row
operations.
Row-echelon form (REF) and reduced row-echelon form (RREF)
7/26/2019 UMEP Sample
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UMEP Mathematics Exam Revision Book
Copyright Victor Lin 2014 Eigenvalues and Eigenvectors 70
To diagonalize an n n matrix A :1. Find the eigenvalues 1 2, , ,
n .
2. Find the eigenvectors1 2, , ,
nv v v that correspond to those eigenvalues.
3. Pis the n n matrix whose columns are the eigenvectors, 1 2[
]nP v v v .
4.
D is the n n diagonal matrix whose main diagonal entries are the
corresponding
eigenvalues,
1
2
3
0 0 0
0 0 0
0 0 0
0 0n
D
.
Powers of a Matrix
Once we have diagonalized the matrix A , then we can easily
findk
A . If1
D P AP
, then1A PDP
. Therefore:
1 1 1 1 1( )k k kA PDP PDP PDP PDP PD P
We can calculatekD easily because if
1
2
3
0 0 0
0 0 0
0 0 0
0 0 n
D
, then
1
2
3
0 0 0
0 0 0
0 0 0
0 0
k
k
k k
k
n
D
.
This is very useful for Markov Chain problems.
Diagonalization of Symmetric Matrices
An n n matrix Q is orthogonal if and only if its columns form an
orthonormal basis for n .
To determine whether a matrix Q is orthogonal, check whether1 TQ
Q .
An n n matrix A is orthogonally diagonalizable if there exists
orthogonal Q such thatTQ AQ D . This is nothing special. We have
simply replaced Pwith Q . The difference is this:
whereas Pis made up of eigenvectors, Q is made up of
eigenvectors after Gram-Schmidt
procedure has been applied.
A matrix A issymmetric ifT
A A . If A is a real, symmetric matrix, then A is
orthogonallydiagonalizable. The following are also true:
