Chapter 10 Review: Matrix
Algebra
Introduction to MATLAB 7
Engineering 161
Matrix Computations
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In Chapter 9 we are going to explore more about matrix
computations using MATLAB.
Concepts like the transpose, dot product, matrix multiplication,
determinants, and the inverse of a matrix help us solve sets of
simultaneous equations often encountered in engineering.
Additionally we’ll explore other built in functions that help us
construct and manipulate matrices.
We’ll work some problems manipulating matrices and I’ll show
you a program I wrote to solve a set simultaneous equations
encounter in circuit analysis that makes use of the control
structures we studied in Chapter 8 and matrix computations.
The Transpose Operator
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The transpose of a matrix is a new matrix in which the rows of the
original matrix become the columns of the new matrix. Normally in
mathematics the symbol “T” is used to designate the transpose
operator, in MATLAB we use the single quote, “ ' “.
We’ve already seen the use of the transpose operator in some of our
first assignments, suppose
>> x = [ 1, 2, 3];
>> y = x.^2
>> [x’, y’]
prints the table
1 1
2 4
3 9
Here the matrix [x’,y’] has two columns, the first being x’ and the second being
y’.
The Transpose Operator II
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More generally, for some arbitrary m x n matrix A,
A' is a n x m matrix. For example,
>>A = [ 1, 2, 3; 4, 5, 6]; %2 rows, 3 columns
>>A'
1 4
2 5
3 6
Whereas A is a 2 x 3 matrix, A' is a 3 x 2 matrix. The
(i,j) element of A becomes the (j,i) element of A'.
Dot Product
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The Dot Product operator is often used in manipulating vectors in
calculus and physics as well as engineering. This operator operates
on two row or column vectors and computes a scalar quantity.
>> A = [ 1, 2, 3 ];
>> B = [ -1, 0, 1];
>> dot (A,B) = 1* (-1) + 2*0 + 3*1 = 2
or equivalently dot (A,B) = sum (A.*B)
In engineering, the length of a vector L is given by the expression
| L | = sqrt (dot (A,A)). sum (A.^2) is the length squared.
For two vectors, A and B, the cosine of the angle between A and B is
given by the dot product of A and B divided by the length of A and the
length of B, namely, cos (angle) = dot (A,B)/ |A|*|B|
Matrix Multiplication
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Matrix multiplication A*B is key for the process of
solving simultaneous equations, so we need to
understand how it works and when it is defined. It is
defined only when the number of columns of A
equals the number of rows of B, or
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If A is a 2 x 3 matrix and B is a 3 x 4 matrix, then the result written
mathematically as AB is a 2 x 4, i.e., a 2 row and 4 column matrix.
In MATLAB we would compute this new matrix with the command
A*B. Note there is no “ . “ involved with the operator.
If A is a 2 x 3 matrix and C is a 4 x 2 matrix, matrix
multiplication is not defined. (In both of these cases, array
multiplication, .*, is not defined. Why?)
Matrix Multiplication II
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When the multiplication is defined, the element ci,j in the new
matrix is computed by calculating the dot product of row i of A
with column j of B. This operation must be performed for each
element ci,j in the new matrix on the appropriate row and
column.
In MATLAB we’d write C = A*B to carry out this operation. With
A and B being matrices conformable for multiplication, MATLAB
knows what to do.
Just because A*B is defined does not mean that B*A is defined.
When A and B are square matrices of the same order, then A*B
and B*A are both defined. Note that A*B need not be equal to
B*A.
We’ll see in a moment where matrix multiplication comes into
play when solving sets of simultaneous equations.
Matrix Powers
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Square matrices, an n x n matrix A, can be raised to
a power by using the ^ operator. Note this is
different from the array exponentiation we discussed
earlier, where each element of A is raised to a power
using the .^ operator.
Raising a matrix to a power involves matrix
multiplication and hence only applies to square
matrices. Here
>> C = A^2
% A^2 ≠ A.*2
means that C is equal to A multiplied by itself.
Similarly, C = A^3 is equivalent to C = A*A*A
Matrix Inverse
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Remember from mathematics that for two functions, f(x) and
g(x) we say that g(x) is the inverse of f(x) if f(x) * g(x) = 1,
similarly, f(x) is the inverse of g(x) if f(x) * g(x) = 1.
For a square matrix A, if A*B = I and B*A = I where I is the
identity matrix, ones along the diagonal and zeros everywhere
else, then B is the inverse of A.
To compute the inverse of A, simply write B = inv(A)
Many square matrices A do not have inverses, they are called
singular. Use the built in functions det(A) or rank(A) to
determine if the square matrix A has an inverse before trying to
use it in calculations. (More on this topic in a moment.)
Determinants
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A determinant is a scalar computed from the entries of a square
matrix. It is used in the process of computing a matrix’s inverse
and other interesting “things.”
When you study linear algebra you will discover that not all
square matrices have inverses. For the inverse of a square
matrix to exist, the determinant of the matrix must be non-zero.
A zero determinant tells us a number of things about a square
matrix, again things you will learn in linear algebra.
For our purposes, for a square matrix A of any order, MATLAB
provides the built in function det to calculate the determinant of
A,
>> det (A)
returns a scalar, the determinant of A.
Cross Products
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Cross product sometimes call vector product operates
on two vectors (row vectors) and produces a third
vector that is perpendicular to the plane formed by
the original two vectors.
For example, consider the two vectors in the x-y
plane, A = [1 2 0] and B = [3 4 0], then
C = cross(A,B) is a third vector given by C = [0 0 -2]
Cross product and dot product are two operations
commonly used in physics and electrical engineering.
Solutions to Systems of Linear Equations
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Sets of simultaneous equations or systems of linear equations
occur often in engineering problem solving. Consider the
following set of linear equations;
3x + 2y
-z = 10
-x +3y + 2z = 5
x -y -z = -1
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Define two matrices A and B as
>> A = [ 3, 2, -1; -1, 3, 2; 1, -1, -1]; % coefficient’s matrix
>> B = [ 10; 5; 1]
% column vector
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Note A is a 3 x 3 matrix and B is a 3 x 1 column vector.
Solutions to Systems of Linear Equations II
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In linear algebra we would define a third matrix, X as X = [ x; y;
z] a column vector of the unknowns and we’d write the
relationship that AX = B. Convince yourself that A*X = B
following the rules of matrix multiplications reproduces the set
of linear equations that we started with.
To solve this set of equations using linear algebra we need first
to find the inverse of A. The inverse of a matrix is another
matrix such that when you multiple the inverse of A and A
together you get the identity matrix, a special matrix with 1’s
along the diagonal and zeros everywhere else. The identity
matrix is special because when you multiple any matrix by the
identity matrix, it reproduces the original matrix. To make a
long story short, consider the following;
Solutions of Systems of Linear Equations III
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Continuing,
A*X = B
inv (A) * A *X = inv (A) * B or
I *X = inv (A) * B or
X = inv (A) * B
So to compute the solution for x, y, and z, we need to find the
inverse of the matrix A and multiple it by B.
In MATLAB, to compute the inverse of a square matrix A, use
the inv (A) command. Then to compute X we’d enter the
statement
>> X = inv(A)*B
where X will be a 3 element column vector in this case. It is as
easy as that.
Solutions to Systems of Linear Equations IV
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Remember we said that not all square matrices have inverses.
We require that determinant be non-zero. An equivalent
condition is that all the equations in the set of linear equations
be independent, i.e., you can not derive one or more of the
equations from linear combinations of the others.
This is equivalent to saying that the rank (A) = n, the order of
A, i.e., for A an n x n matrix.
Use the MATLAB command rank (A) to see that all the
equations are independent before trying to compute the inv (A),
if not you’ll get an error message from MATLAB.
When rank (A) < n, we say the matrix is singular and hence
doesn’t have an inverse. This also means that the solution to
the set of linear equations can not be determined uniquely.
Solutions to Systems of Linear Equations V
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MATLAB provides a second way to solve sets of equations like
this that is generally faster than computing the inverse
especially for large matrices.
It is called Gaussian elimination. To use this method simply
write following MATLAB expression
>> X = A\B
Note that we are using the forward slanted bar (called matrix left
division) not the normal division bar “ / “. The forward slanted bar
tells MATLAB to perform Gaussian elimination on the matrices A,B to
compute the column vector X. The notation may seem strange until
you study linear algebra, then the notation will seem obvious.
Special Matrices
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MATLAB contains a set of special functions that generate some
special matrices. We’ll look at some of these.
The zeros function generates a matrix containing all zeroes;
>> A = zeros (3); % A is a 3X3 matrix of all 0’s
>> B = zeros (2,3); % B is a 2X3 matrix of all 0’s
To determine the size of a matrix, use the size command. For
example,
>>[m,n] = size (B);
would return two values, m = 2 and n = 3.
>> D = zeros (size (B));
creates a 2 x 3 matrix D of all zeros.
Special Matrices II
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Similarly for the ones function. Here, ones(2) would create a
2X2 square matrix of all ones, while the command ones(3,4)
would create a matrix of 3 rows and 4 columns of all ones.
The eye command is used to create diagonal matrices. eye(3)
would create a 3X3 square matrix with ones along the diagonal
and zeroes everywhere else. We’ve called this the identity
matrix previously. eye(2,3) would create a matrix of 2 rows and
3 columns, element 1,1 and 2,2 would be set equal to 1 and
zero everywhere else.
A 3x3 identity matrix I is created by I = eye(3)
Special Matrices III
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The diag function can be used to either define a
diagonal matrix or extract the diagonal from a matrix.
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For B = [1, 2, 3] a row vector, A = diag (B) creates a 3X3
matrix A with the elements along the diagonal being the
elements from B.
If A is a square matrix, then C = diag (A) extracts the
elements along the diagonal of A and creates the row vector
C.
You can pass a second parameter to the diag function, an
integer to tell the function to choose a diagonal other than
the main diagonal. k = 1 tells the function to choose the
diagonal above the main diagonal, while k = -1 specifies the
one below the main diagonal. Consider the example,
Special Matrices IV
>>B = [1,2,3,4;5,6,7,8;9,10,11,12;13,14,15,16]
B=
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
>>C = diag(B,1)
C=
2 7 12
By specifying k other than 1 or -1, you can selected arbitrary
diagonals above or below the main diagonal.
Other Matrices
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MATLAB contains build in functions that will generate
matrices that are useful for testing numerical
techniques, that serve in computational algorithms or
that are just plain interesting, consider
pascal(n)
magic(n)
v= [1 2 3];
vander(v)
Chapter 9 Assignments
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These assignments let you practice the
concepts discussed in Chapter 9.
Assignments 10.18, 10.20
Note: Chp. 11 is about Data Types. You
might want to read over the material on
floating point, integers and complex numbers
to acquaint yourself to these other data
types.