From our understanding of
eigenvalues and eigenvectors we have
discovered several things about our operator matrix,
. We know that if the
eigenvectors of
span
and we know how to express any vector
in terms of
, then we have the operator
all figured out. If we have
acting on
, then this is equal to
acting on the combinations of
eigenvectors. Which we know proves to be fairly easy!
We are still left with two questions that need to be
addressed:
When do the eigenvectors
of
span
(assuming
are linearly independent)?
How do we express a given vector
in terms of
?
Answer to question #1
When do the eigenvectors
of
span
?
If
has
distinct eigenvalues
where
and
are integers, then
has
linearly independent
eigenvectors
which then span
.
The proof of this statement is not very hard, but is not
really interesting enough to include here. If you wish toresearch this idea further, read Strang, G., "Linear Algebra
and its Application" for the proof.
Furthermore,
distinct
eigenvalues means
has
distinct roots.
Answer to question #2
How do we express a given vector
in terms of
?
We want to find
such that
In order to find this set of variables, we will begin by
collecting the vectors
as columns in a n×n matrix
.
Now
[link] becomes
or
which gives us an easy form to solve for our variables inquestion,
:
Note that
is invertible since
it has
linearly independent
columns.
Aside
Let us recall our knowledge of functions and their basis and
examine the role of
.
where
is
just
expressed
in a different
basis :
transforms
from the
standard basis to the basis
Matrix diagonalization and output
We can also use the vectors
to represent the output,
, of a system:
where
is the matrix
with the eigenvalues down the diagonal:
Finally, we can cancel out the
and are left with a final equation for
:
Interpretation
For our interpretation, recall our key formulas:
We can interpret operating on
with
as:
where the three steps (arrows) in the above illustration represent
the following three operations:
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.