Eigenvalues of a matrix. Hello friends, today it’s all about the eigenvalues of a matrix. Have a look!!
If you’re looking for more in eigenvalues and eigenvectors of matrices, do check-in:
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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. And any matrix is a square matrix if the number of rows and the number of columns of that matrix are the same.
Now I will give an example.
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:
Now here the given coefficient matrix is
And I have to find out its eigenvalues and the eigenvector corresponding to each eigenvalue.
Now 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.
Now, to get the eigenvalues of any matrix, first of all, I’ll get the characteristic equation of the matrix. So, I’ll start with that.
As I have already mentioned above, 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.
And that means I have to expand the determinant.
Therefore it will be
Next, I’ll evaluate it. So what I get is
Now I’ll simplify it. And that gives
As I can see, is a common term through out the expression. So I can say that the value of will be
And that gives
Next, I’ll factorise to get
Thus the characteristic equation of the matrix is
Next, I’ll 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.
Now the second part of this example is in my other post namely eigenvectors of a matrix.
Dear friends, this is the end of 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!!