Eigenvalues of a matrix. Hello friends, today it’s all about the eigenvalues of a matrix. Have a look!!
Eigenvalues of a matrix
Suppose I have a matrix, say . Also, it’s a matrix.
That means it has 3 rows and 3 columns.
Now if is an eigenvalue of the matrix , then it will satisfy an equation
This equation is called the characteristic equation of the matrix .
Here is a matrix. Therefore will have 3 values.
If it would have been matrix or matrix, then would have 4 or 5 values respectively.
Note: It is only possible to get the eigenvalues of any matrix if it’s a square matrix.
Now I will give an example.
If interested, you can also check out other posts on eigenvalues and eigenvectors such as eigenvectors of a matrix, eigenvectors of repeated eigenvalues, complex eigenvalues and eigenvectors of a matrix and so on.
Example of the eigenvalues of a matrix
Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.
So here is the example.
According to Stroud and Booth (2013)* “For the coefficient matrix …, determine the eigenvalues and an eigenvector corresponding to each eigenvalue:
Today I’ll only do the first part of the problem, that is, to get the eigenvalues of a matrix.
I have already shown the second part, that is, the corresponding eigenvectors on another post.
Here the given matrix is
Now, to get the eigenvalues of any matrix, the first task is to get its characteristic equation. So, I’ll start with that.
I have already mentioned above that the characteristic equation of any matrix is
Here is the eigenvalue of the matrix . is the unit matrix or the identity matrix.
Now means the determinant of the matrix
Thus the matrix is
Therefore the characteristic equation of the matrix is
Now I will simplify the left-hand side of the equation.
That means I’ll expand the determinant.
Therefore it will be
Next, I’ll evaluate it to find
So I’ll simplify it a bit more to get
Thus the characteristic equation of the matrix is
Now the next step is to get the eigenvalues of this matrix.
As I know it already, the eigenvalues of any matrix are the solutions of the characteristic equation of the matrix.
Now in this example, the characteristic equation is
Thus the solutions are
Hence I can conclude that the eigenvalues of the given matrix are and respectively.
This is the first part of the example.
You will find the second part on my other post in eigenvectors of a matrix.
Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!