Eigenvalues and Eigenvectors play a critical role
in determining observable measurements of quantum systems,
in evaluating the stability of mechanical structures,
in analyzing the feedback loops of electric circuits,
and in many other areas.
To fully understand physics and engineering,
it is necessary to understand Eigenvalues and Eigenvectors.
Suppose we have an object.
Here we are viewing the object from different perspectives.
Now, suppose we apply what we call a linear transformation.
As we again view the object from different perspectives,
we see that the object is now distorted.
Let us consider the points on this object before the transformation.
Each of these points can be represented by an arrow.
When the transformation is applied, the length and direction of each arrow can change.
There are some arrows that
point along the same line both before and after the transformation.
For example, these three arrows will simply
have their lengths multiplied by a constant.
The length of the orange arrow is multiplied by negative one.
The length of the blue arrow is multiplied by two.
And the length of the green arrow is multiplied by one half.
We say that negative one, two, and one half
are the “eigenvalues” of this transformation.
The arrows that point along the same line both before and after the transformation
are what we call the “eigenvectors.”
The amounts by which the lengths of each of these arrows is multiplied
are what we call the eigenvalues.
The eigenvalues and eigenvectors
dictate the nature of the transformation for the entire object.
Consider, for example, the point symbolized by this white arrow.
The white arrow is not aligned with the directions of any of the eigenvectors,
which are symbolized by the orange, green, and blue lines.
But, the white arrow can be thought of as a combination of arrows
that are parallel to the directions of these three lines.
When the transformation is applied, the lengths of each of these arrows
is multiplied by the eigenvalue associated with that direction.
That is, the orange, green, and blue arrows continue pointing in the same direction,
but are multiplied by “-1”, “1/2”, and “2.”
In this way, the transformation of any point on this object
can be determined by using the eigenvalues and eigenvectors.
The eigenvectors do not necessarily have to be 90 degrees to one another.
For example, let us consider this new object, and this new transformation.
The red and green arrows are eigenvectors of this transformation.
The green arrow is multiplied by one, and the red arrow is multiplied by two.
Now consider the transformation of this white arrow.
The white arrow can be thought of as the
combination of arrows parallel to the green and red lines.
During the transformation, this green arrow is multiplied by one,
and this red arrow is multiplied by two.
The eigenvalues and eigenvectors do not necessarily have to consist of real numbers.
The eigenvalues of this transformation are the imaginary numbers “i” and negative “i.”
And the eigenvectors are the following.
Let us say that the red arrow is signified by a “1” followed by a “0.”
And let’s say that the yellow arrow is signified by a “zero” followed by a “one.”
Before the transformation,
the blue arrow is in the same direction as the red arrow, but double its length.
Therefore, using the notation described, the blue arrow before the transformation
would be signified by a “2” followed by a “0.”
After the transformation, the blue arrow is pointed in the
opposite direction of the yellow arrow, and double its length.
Therefore, the blue arrow after the transformation
would be signified by a “0” followed by a “-2.”
Let us consider the blue arrow before the transformation.
The blue arrow can be thought of as the sum of the two eigenvectors.
Suppose we multiply each eigenvector by its corresponding eigenvalue.
The result is the new direction of our blue arrow after the transformation.
Therefore, everything discussed here is still true
even for imaginary values for eigenvalues and eigenvectors.
Much more information about linear transformations is available in the video
“Linear Algebra – Matrix Transformations.”
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