Multiplying a matrix and a vector is a
special case of matrix multiplication. Circuit equations and state equations
representing linear system dynamics contain products of a matrix and a vector.
In the first lesson on circuit analysis, equations that come about by writing
node equations can be put into a vector-matrix representation that includes a
term that is a matrix - the conductance matrix - multiplied by a vector - the
vector of node voltages.
Since vector-matrix representations are
encountered often in electrical engineering, you need to be very familiar with
basic operations. In this lesson, we will examine mutiplying a matrix and a
vector. In the basic lesson on circuits, we encountered this vector-matrix
representation for the circuit below.
The form we are interested in is this. We
want to be able to evaluate a matrix vector product of this form whenever we
encounter one.
The algorithm for computing the product is
best presented visually. Here it is.
There are some things to remember about
matrix-vector multiplication.
The matrix is assumed to be N
x M. In other words:
The matrix has N rows.
The matrix has M columns.
For example, a 2 x 3
matrix has 2 rows and 3 columns.
In matrix-vector
multiplication, if the matrix is N x M, then the vector must have a
dimension, M.
In other words, the
vector will have M entries.
If the matrix is 2 x 3,
then the vector must be 3 dimensional.
This is usually stated as
saying the matrix and vector must be conformable.
Then, if the matrix and
vector are conformable, the product of the matrix and the vector is a
resultant vector that has a dimension of N. (So, the result could be a
different size than the original vector!)
For example, if the
matrix is 2 x 3, and the vector is 3 dimensional, the result of the
multiplication would be a vector of 2 dimensions.
It is possible to express the calculations
mathematically.
Let the matrix be represented
by A.
The elements of A are aij,
where,
i is the row index and
takes on values from 1 to M.
j is the column index and
takes on values from 1 to N.
Let the vector be represented
by b.
The elements of b
are bj, where,
j is the index and takes
on values from 1 to N.
The product is c = A*b,
The product is a vector
of length M.
Then, the calculation for the the terms in
the product vector are given by:
This expression just puts the process for
calculating the product into standard mathematical form. What it says to do is
the following.
To calculate the jth
entry in the product vector.
Multiply entries in the jth
row of the matrix, A, by the corresponding entries in the vector, b,
and sum all of the terms.
So, now you should be able to perform these
calculations. Let's look at some example problems.
