Download Refresher: Vector and Matrix Algebera - FAMU

Refresher:
Vector and Matrix Algebra
Mike Kirkpatrick
Department of Chemical Engineering
FAMU-FSU College of Engineering
Outline

Basics:
• Operations on vectors and matrices

Linear systems of algebraic equations
•
•
•
•

Gauss elimination
Matrix rank, existence of a solution
Inverse of a matrix
Determinants
Eigenvalues and Eigenvectors
• applications
• diagonalization
more
Outline cont’

Special matrix properties
• symmetric, skew-symmetric, and
orthogonal matrices
• Hermitian, skew-Hermitian, and unitary
matrices
Matrices



A matrix is a rectangular array of numbers (or
functions).
a11 a12  a1n 
a

a
 21 22






amn 
am1
The matrix shown above is of size mxn. Note that
this designates first the number of rows, then the
number of columns.
The elements of a matrix, here represented by
the letter ‘a’ with subscripts, can consist of
numbers, variables, or functions of variables.
Vectors


A vector is simply a matrix with either one row or
one column. A matrix with one row is called a row
vector, and a matrix with one column is called a
column vector.
Transpose: A row vector can be changed into a
column vector and vice-versa by taking the
transpose of that vector. e.g.:
if
A  3 4 5
3 
then AT  4
5 
Matrix Addition


Matrix addition is only possible between
two matrices which have the same size.
The operation is done simply by adding
the corresponding elements. e.g.:
1 3  6 2 7 5
4 7  3 1   7 8

 
 

Matrix scalar multiplication

Multiplication of a matrix or a vector
by a scalar is also straightforward:
1 3  5 15 
5* 



4 7 20 35
Transpose of a matrix


Taking the transpose of a matrix is
to that of a vector:
1 3 8 
1 4
if A  4 7 2, then AT  3 7
6 5 0 
8 2
similar
6
5
0
The diagonal elements in the matrix are
unaffected, but the other elements are
switched. A matrix which is the same as
its own transpose is called symmetric, and
one which is the negative of its own
transpose is called skew-symmetric.
Matrix Multiplication




The multiplication of a matrix into another matrix
not possible for all matrices, and the operation is
not commutative:
AB ≠ BA in general
In order to multiply two matrices, the first matrix
must have the same number of columns as the
second matrix has rows.
So, if one wants to solve for C=AB, then the
matrix A must have as many columns as the
matrix B has rows.
The resulting matrix C will have the same
number of rows as did A and the same number of
columns as did B.
Matrix Multiplication

The operation is done as follows:
using index notation: C 
jk


n
l 1
Ajl Blk
for example:
4 3 
 4 * 2  3 *1 4 * 5  3 * 6 
 2 5 



AB  7 2 
  7 * 2  2 *1 7 * 5  2 * 6 

1 6

9 0 
9 * 2  0 *1 9 * 5  0 * 6 
11 38 
 16 47 
18 45 
Linear systems of equations



One of the most important application of
matrices is for solving linear systems of
equations which appear in many different
problems including electrical networks,
statistics, and numerical methods for
differential equations.
A linear system of equations can be written:
a11x1 + … + a1nxn = b1
a21x1 + … + a2nxn = b2
:
am1x1 + … + amnxn = bm
This is a system of m equations and n
unknowns.
Linear systems cont’


The system of equations shown on the
previous slide can be written more
compactly as a matrix equation:
Ax=b
where the matrix A contains all the
coefficients of the unknown variables from
the LHS, x is the vector of unknowns, and
b a vector containing the numbers from
the RHS
Gauss elimination


Although these types of problems can be
solved easily using a wide number of
computational packages, the principle of
Gaussian elimination should be
understood.
The principle is to successively eliminate
variables from the equations until the
system is in ‘triangular’ form, that is, the
matrix A will contain all zeros below the
diagonal.
Gauss elimination cont’
A very simple example:
-x + 2y = 4
3x + 4y =38
first, divide the second equation by -2, then
add to the first equation to eliminate y;
the resulting system is:
-x + 2y = 4
-2.5x = -15 
x=6

y=5

Matrix rank



The rank of a matrix is simply the number
of independent row vectors in that matrix.
The transpose of a matrix has the same
rank as the original matrix.
To find the rank of a matrix by hand, use
Gauss elimination and the linearly
dependant row vectors will fall out,
leaving only the linearly independent
vectors, the number of which is the rank.
Matrix inverse




The inverse of the matrix A is denoted as A-1
By definition, AA-1 = A-1A = I, where I is the
identity matrix.
Theorem: The inverse of an nxn matrix A
exists if and only if the rank A = n.
Gauss-Jordan elimination can be used to find
the inverse of a matrix by hand.
Determinants



Determinants are useful in eigenvalue
problems and differential equations.
Can be found only for square matrices.
Simple example: 2nd order determinant
1 3
det A 
 1* 7  3 * 4  5
4 7
3rd order determinant

The determinant of a 3X3 matrix is
found as follows:
a11 a12
det A  a21 a22
a31 a32

a13
a23  a11
a33
a22
a23
a32
a33
 a12
a21 a23
a31 a33
 a13
a21 a22
a31 a32
The terms on the RHS can be
evaluated as shown for a 2nd order
determinant.
Some theorems for determinants


Cramer’s: If the determinant of a
system of n equations with n
unknowns is nonzero, that system
has precisely one solution.
det(AB)=det(BA)=det(A)det(B)
Eigenvalues and Eigenvectors



Let A be an nxn matrix and consider the
vector equation:
Ax = x
A value of  for which this equation has a
solution x≠0 is called an eigenvalue of the
matrix A.
The corresponding solutions x are called
the eigenvectors of the matrix A.
Solving for eigenvalues
Ax=x
Ax - x = 0
(A- I)x = 0



This is a homogeneous linear system,
homogeneous meaning that the RHS are
all zeros.
For such a system, a theorem states that
a solution exists given that det(A- I)=0.
The eigenvalues are found by solving the
above equation.
Solving for eigenvalues cont’

Simple example: find the eigenvalues for
the matrix:
 5 2 
A

2

2



Eigenvalues are given by the equation
det(A-I) = 0:
5
2
det( A   I ) 
2
2
 (5   )( 2   )  4   2  7  6

So, the roots of the last equation are -1
and -6. These are the eigenvalues of
matrix A.
Eigenvectors



For each eigenvalue, , there is a
corresponding eigenvector, x.
This vector can be found by substituting
one of the eigenvalues back into the
original equation: Ax = x : for the
example:
-5x1 + 2x2 = x1
2x1 – 2x2 = x2
Using =-1, we get x2 = 2x1, and by
arbitrarily choosing x1 = 1, the eigenvector
corresponding to =-1 is:
1 
x1   
 2
and similarly,
2
x2   
 1
Special matrices



A matrix is called symmetric if:
AT = A
A skew-symmetric matrix is one for
which:
AT = -A
An orthogonal matrix is one whose
transpose is also its inverse:
AT = A-1
Complex matrices


If a matrix contains complex (imaginary)
elements, it is often useful to take its
complex conjugate. The notation used for
the complex conjugate of a matrix A is: 
Some special complex matrices are as
follows:
Hermitian:
Skew-Hermitian:
Unitary:
T = A
T = -A
T = A-1