9.1 Matrix Multiplication
9.2 Determinants and Matrix Division
We will expand on the use of matrix operations on arrays in this
chapter. These operations give MATLAB (for Matrix Laboratory) its
name. As we saw in the section on element by element operations,
there are many cases where MATLAB thinks we want to do matrix
operations unless we specify (with the use of a period) otherwise.
First a trivial, but important operation: forming the transpose. This
is done in MATLAB with the single quote: '. It essentially reverses
rows and columns in a matrix. If the matrix is complex in MATLAB it
also takes the conjugate of each element. If we want to keep from
getting a conjugate, we need to insert a period before the quote.
Here are some examples:
>> disp([1 2]')
1
2
>> a=[1 2 3
4 5 6];
>> disp(a')
1 4
2 5
3 6
>> z=a+i*[-1 0 1;-2 1 2];
>> disp(z)
1.0000 - 1.0000i 2.0000 3.0000 + 1.0000i
4.0000 - 2.0000i 5.0000 + 1.0000i 6.0000 + 2.0000i
>> disp(z')
1.0000 + 1.0000i 4.0000 + 2.0000i
2.0000 5.0000 - 1.0000i
3.0000 - 1.0000i 6.0000 - 2.0000i
>> disp(z.')
1.0000 - 1.0000i 4.0000 - 2.0000i
2.0000 5.0000 + 1.0000i
3.0000 + 1.0000i 6.0000 + 2.0000i
9.1 Matrix Multiplication
Matrix multiplication finds extensive applications in a number of areas of
mathematics and solution of equations. Students are usually
introduced to it in the solution of $N$ simultaneous linear equations
in $N$ unknowns. If the equations are written:
a11x1 + a12x2 + ... + a1NxN = b1 a21x1 + a22x2 + ... + a2NxN = b2 . . . . aN1x1 + aN2x2 + ... + aNNxN = bN
we find that they express what we want to solve, but they are
cumbersome to write out each time we look at another problem. Looking
carefully at this and similar problems, we see that there are three
"entities" that we will deal with.
The set of unknowns: x1, x2, ..., xN. The set of coefficients
multiplying the unknowns:
a11 a12 ... a1N a21 a22 ... a2N . . . . aN1 aN2 ... aNN
The set of given elements: b1, b2, ..., bN
It would be a lot easier to call these x, A and b and to write our
set of equations as:
Ax = b.
We can do just that if:
1) We write our "vectors" x and b as matrices with a single
column,
2) We define matrix multiplication of A (with elements: aij) by
B (with elements: bjk) to give C (with elements
cik) such that: if A is an M row by N column matrix and B
is an N row by P column matrix, then AB = means that:
giving
C , an M row by P column matrix
3) We understand that two matrices are equal if and only if all
elements in like positions are equal.
Two very important points about this operation:
a) Arrays A and B can be multiplied together in the matrix sense only if the number of columns in the first array is equal to the number of rows in the second array.
b) The product: AB is nearly always not equal to BA even when both products are defined.
Many of the error messages that you have seen in these notes (and probably in your practice sessions) when you tried to multiply arrays together occur as a result of the violation of the first of these points. Here is a MATLAB session to illustrate the first point:
>> a=[1 2 3
4 5 6];
>> a*[1 2]
??? Error using ==> *
Inner matrix dimensions must agree.
>> a*[1 2 3]
??? Error using ==> *
Inner matrix dimensions must agree.
>> disp(a*[1 2]')
??? Error using ==> *
Inner matrix dimensions must agree.
>> disp(a*[1 2 3]')
14
32
The matrix a has three columns, therefore it can multiply only matrices with three rows. Row vectors of any length have only one row. A column vector of exactly three rows is the only one that worked. Here is a typical case that shows how matrix multiply depends on the order in which the matrices are listed:
>> b=[1 2
3 4
5 6];
>> disp(a*b)
22 28
49 64
>> disp(b*a)
9 12 15
19 26 33
29 40 51
Matrix multiply gives a direct way to find the vector product of
two vectors of the same length or of a vector with itself. For real
vectors with N elements, this is defined as: 
It is given in MATLAB by:
>> x=[1 2 3];
>> disp(x*x') <-- Note the transpose using a quote
14
>> y=[-1 0 1];
>> disp(x*y') <-- Note the transpose using a quote
2
9.2 Determinants and Matrix Division
MATLAB gives us two matrix division operations. We will consider only
the one that solves the set of equations: AX=B for X, given A and B.
This is done with:
>> X=A\B
The simplest case is the one where the number of unknowns equals the number of equations. In that case and if the determinant of A is not 0 (or not so small that the computer can not find one divided by it), then there should be a unique solution to the set of equations. This is frequently represented by:
A-1B
where A-1 is defined as the inverse matrix to A .The
matrix product of the inverse of A with B then gives our desired
solution.
Let's see what can be done in MATLAB to demonstrate this solution
procedure. Suppose we want to solve:
3x1 + x2 - 2x3 = 1 x1 + 2x2 - x3 = 0 -2x1 - x3 = 5
We set up our matrix a:
>> a=[3 1 -2
1 2 -1
-2 0 -1];
The determinant can be found with det:
>> disp(det(a)) -11
It is a long way from zero, so we expect a unique solution.
>> b=[1 0 5]'; >> disp(a\b) -1.1818 -0.7273 -2.6364
Now let's check that answer (conveniently stored in ans):
>> disp(a*ans)
1.0000
0
5.0000
We could also have carried out the solution using the inverse matrix function: inv. Here is how that would have worked:
>> ai=inv(a);
>> disp(ai)
0.1818 -0.0909 -0.2727
-0.2727 0.6364 -0.0909
-0.3636 0.1818 -0.4545
>> disp(ai*b)
-1.1818
-0.7273
-2.6364
We arrived at the same answer as before. Since matrix inversion
takes many more operations than the solution of the same number of
equations, we would never follow this route unless we had the same
set of equations to solve (with different values for b) many
times.
Our main restriction on use of B\A is that the number of rows in each
matrix must be the same. In most cases where the number of equations
is larger than the number of unknowns; i.e. the number of rows of A
is larger than the number of columns, there is no solution that will
satisfy all the equations. In that case, we say that the system of
equations is over specified. MATLAB then finds the best solution it
can by making the sum of the squares of the errors in the equations
as small as possible. This is said to give the ``least square error"
solution. An example is:
4x1 + 2x2 = 3 x1 - x2 = -1 3x1 + 4x2 = 7
MATLAB gives the following for the two unknowns that come closest to satisfying all three equations:
>> a=[4 2
1 -1
3 4];
>> b=[3 -1 7]';
>> disp(a\b)
0.0378
1.6324
If we multiply the matrix by that result, we get:
>> disp(a*[.0378;1.6324])
3.4160
-1.5946
6.6430
This procedure finds frequent use in generalized curve
fitting.
Finally, if there are more unknowns than there are equations (or if
the matrix is square but singular) then there are an infinity of
solutions. Here, MATLAB gives us one such solution. A singular
example is shown next:
>> a=[ 1 2 3
4 3 2
5 5 5];
>> disp(det(a))
0
>> disp(a\[1 2 3]')
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 1.110223e-17
0.0250
0.7500
-0.1750
The answer given works well when we multiply it by a:
>> disp(a*[.025 .75 -.175]')
1
2
3
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