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Multiplying
A Matrix And A Vector
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. (Click
here to go to that point in the lessons where that is presented.)
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
a_{ij}, 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 b_{j}, 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 j^{th}
entry in the product vector.
Multiply entries in the
j^{th} 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.
Questions
Q1. For
this matrix and vector, is the product defined?
Q2. For
this matrix and vector, is the product defined?
Problems
P1 In
this matrix-vector product, the result is a vector, c,
with two components, c_{1}
and c_{2}.
Calculate the components of the product and enter your answer in the spaces
below.
First, calculate the value
of c_{1}.
P1 Next,
calculate the value of c_{2}.
Finally, we have a calculator you can use to avoid doing these kinds
of problems by hand. Here is the calculator. Here's how to
use the calculator.
Determine the size of
the matrix and enter N (#rows) and M (#columns).
The maximum matrix size
is 5 x 5.
Create a matrix and a
vector by clicking the button. You will automatically get a conformable
matrix and vector.