Introduction
Similar matrices represent the same linear map under two (possibly) different bases. It is the fundamental for matrix diagonalization.
In this blog post, I would like to quickly discuss how to understand similar matrices.
Definition of Similar Matrices
Two matrices and are similar if there exists an invertible matrix such that .
The matrix is invertible if and only if the columns of , , are linearly independent, i.e., constitutes a basis or span for . Let be a basis for , which is different from the standard basis . Let be a vector in and its coordinates are represented using standard basis for simplicity.
By the definition of similar matrices, we have the vector coordinate transformation .
Recall, because is a basis for , can be written as a linear combination of the basis,
where
and
are the new coordinates for using the basis.
Because is invertible, we have
Thus, to transform the vector coordinates from the standard basis coordinate system to the new basis coordinate system, we do
To transform the vector coordinates from the new basis coordinate system to the standard basis coordinate system, because the matrix is invertible,
The vector coordinate transformation using similar matrices implies that the vector coordinate transformation in the standard basis coordinate system is equivalent as transforming the vector coordinates from the standard basis coordinate system to the new basis coordinate system, performing transformation in the new basis coordinate system, and transforming the vector coordinates from the new basis coordinate system back to the standard basis coordinate system.
The transformation between the transformation matrix and the transformation is similar in a way that the coordinates of the vector, , in the basis coordinate system, after the transformation and the transformation back to the standard basis coordinate system, is exactly the same as the coordinates of the same vector in the standard basis coordinate system after the transformation . That is to say,
Alternatively, we could also say the transformation between the transformation matrix and the transformation is similar in a way that the coordinates of the vector, , in the basis coordinate system, after the transformation , is exactly the same as the coordinates of the same vector in the standard basis coordinate system after the transformation and the transformation to the basis coordinate system. That is to say,
In fact, similar matrices and represent the same linear map under two (possibly) different bases. To see this, given the linearly independent column vectors from the matrix represented using standard basis coordinates, , their basis coordinates are .
Thus the vector coordinates transformation using the standard basis is just the “same” as the vector coordinates transformation using the basis followed by converting the basis back to the standard basis from the basis. For example, because
Finally, in practice, does not have to be represented using standard basis. That’s to say, it’s not necessary that one of the coordinate systems has to be a standard basis coordinate system.
Eigenvalues of Similar Matrices
Similar matrices have some interesting properties related to eigenvalues and eigenvectors.
Similar Matrices Have the Same Eigenvalues
Similar matrices have the same eigenvalues.
Proof
Suppose matrices and are similar and , the characteristic polynomial for computing the eigenvalues of becomes
Because the characteristic polynomials for computing the eigenvalues of and are exactly the same, similar matrices and have the same eigenvalues.
This concludes the proof.
Eigenvectors of Similar Matrices
Suppose matrices and are similar, . Because similar matrices have the same eigenvalues, we further suppose is the eigenvalue for and , and where is an eigenvector for corresponding to the eigenvalue .
Because of the following relationship,
Therefore, an eigenvector for corresponding to the eigenvalue is .
Suppose where is an eigenvector for corresponding to the eigenvalue , similarly, we could also derive that an eigenvector for corresponding to the eigenvalue is .
Eigenspace of Similar Matrices
Suppose matrices and are similar, . is the eigenvalue for and , the -eigenspace of is the solution set of , i.e., the nullspace of matrix , .
For any eigenvector in the -eigenspace of , is an eigenvector in the -eigenspace of . We could say, takes the -eigenspace of to the -eigenspace of . Similarly, takes the -eigenspace of to the -eigenspace of .
Eigenvectors with Distinct Eigenvalues
Eigenvectors with distinct eigenvalues are linearly independent.
Let be eigenvectors of a matrix , and suppose that the corresponding eigenvalues are distinct. Then is linearly independent.
Proof
We will prove by contradiction.
Suppose were linearly dependent.
This means that we can rearrange the order of , for some , is a span that is linearly independent, and is a linearly combination of the span.
Multiplying both side of the equation by ,
Multiplying both side of the equation by ,
Thus,
We further have
Because , the linear equation
has non-zero solutions.
But is linearly independent and the above linear equation only has the zero solution. which raises a contradiction.
Therefore, is linearly independent.
This concludes the proof.
Notice that this theorem does not assume whether the eigenvectors and eigenvalues are real or complex.
Similar to Diagonal Matrix
Suppose matrices and are similar, , , and is a diagonal matrix.
It’s easy to see and verify that the standard basis is a set of linearly-independent eigenvectors for any diagonal matrix .
Suppose the eigenvalues for each of the eigenvectors are , the -eigenspace of is , the -eigenspace of is , etc.
Because we have derived that takes the -eigenspace of to the -eigenspace of . In this case, takes the -eigenspace of to the -eigenspace of , a set of linearly independent eigenvectors of are , and the -eigenspace of is , the -eigenspace of is , etc.
Because of the following relationship,
we could see that the diagonal matrix essentially scales the in the basis coordinate system by the corresponding eigenvalue . In the standard basis coordinate system, correspondingly, the matrix scales by the corresponding eigenvalue . Both of the operations, even though in different coordinate systems, are the same scaling operation using eigenvalue .
References