To understand how to "find" similar matrices, it's essential to grasp their precise definition and the implications of that definition. In linear algebra, two matrices are considered similar if one can be transformed into the other through a change of basis. This relationship reveals fundamental properties shared between them, even if their entries look different.
Defining Similar Matrices
According to the provided definition:
"Suppose A and B are two square matrices of size n. Then A and B are similar if there exists a nonsingular matrix of size n, S, such that A=S−1BS."
— Similar Matrices, Beezer via AIMath.org
This definition provides the direct criterion for similarity. In essence, to determine if matrices A and B are similar, you must verify the existence of such a non-singular matrix S. If such an S exists, then A and B are similar. If it does not exist, they are not.
The term S⁻¹BS represents a change of basis. If A and B are similar, they represent the same linear transformation, but with respect to different bases. The matrix S acts as the "change of basis matrix" that converts coordinates from one basis to another.
Identifying Similar Matrices: Practical Approaches
While the definition requires finding the matrix S, directly computing S can be challenging. Instead, we often look for invariants—properties that remain unchanged under similarity transformations. If two matrices do not share these invariants, they cannot be similar. If they do share them, they might be similar, and further investigation (like finding S or comparing canonical forms) would be needed.
Here are key properties that similar matrices always share:
| Property | Description |
|---|---|
| Determinant | det(A) = det(B) |
| Trace | tr(A) = tr(B) (sum of diagonal elements) |
| Eigenvalues | A and B have the exact same set of eigenvalues. |
| Characteristic Polynomial | A and B have the same characteristic polynomial. |
| Rank | rank(A) = rank(B) |
| Nullity | nullity(A) = nullity(B) (dimension of the null space) |
| Minimal Polynomial | A and B have the same minimal polynomial. |
How These Properties Help "Find" Similarity:
- Necessary Conditions: Checking these properties is often the first step. If any of these properties differ between two matrices A and B, then A and B are not similar. This acts as a quick filter.
- Sufficient Conditions (with Caveats): If all these properties are the same, it suggests similarity but doesn't guarantee it, except in specific cases (e.g., if both are diagonalizable or if their minimal polynomials are equal and they have the same characteristic polynomial).
Constructing a Similar Matrix
If you have a matrix A and a non-singular matrix S, you can easily construct a matrix B similar to A using the formula derived from the definition:
- Given
A = S⁻¹BS, you can rearrange it to find B:
SAS⁻¹ = S(S⁻¹BS)S⁻¹
SAS⁻¹ = (SS⁻¹)B(SS⁻¹)
SAS⁻¹ = IBI
B = SAS⁻¹
This means that if you choose any invertible matrix S, the resulting matrix B will always be similar to A.
Advanced Methods for Verifying Similarity
For a complete determination of similarity, especially when the simpler property checks are insufficient, more advanced techniques are used:
- Diagonalization: If both matrices A and B can be diagonalized, they are similar if and only if they are similar to the same diagonal matrix (i.e., they have the same eigenvalues, and the dimensions of their eigenspaces match).
- Jordan Canonical Form (JCF): Every square matrix over an algebraically closed field (like complex numbers) is similar to a unique Jordan Canonical Form. Therefore, two matrices are similar if and only if they have the same Jordan Canonical Form. Finding the JCF for a given matrix is a definitive way to determine its similarity class.
In summary, finding similar matrices primarily involves understanding the transformation defined by A = S⁻¹BS. Practically, it means looking for a non-singular matrix S that satisfies this equation, or, more commonly, checking for shared invariant properties that are characteristic of similar matrices.
