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In linear algebra, the concept of matrix similarity or equivalence is very important for understanding how two different matrices can represent the same linear transformation in different spaces. Imagine two different portraits of the same object, but taken from different perspectives.
Two matrices are said to be similar if there exists an invertible matrix such that:
The matrix in this case is called the similarity transformation matrix.
To understand why matrix similarity is so important, we need to look at its relationship with basis transformation. Let be the canonical basis and be another basis of .
If is an invertible matrix with columns :
Then we have or for . The matrix represents the basis transformation.
A vector can be expressed in the canonical basis through coordinates and in the basis through coordinates :
The matrix represents the coordinate transformation.
Now consider the linear transformation . In the canonical basis, is expressed through coordinates , while in the basis through coordinates :
Therefore:
or in other words:
In the basis , the linear transformation is represented by with the matrix:
This is why similar matrices represent the same linear transformation but viewed from different bases. Similar matrices represent the same linear transformation with respect to different bases of .
Similar matrices have several fundamental properties that are very useful. Since they represent the same linear transformation in different spaces, similar matrices preserve the same intrinsic characteristics.
Based on the theorem about similar matrices, if matrices and are similar, then they both have:
For the determinant, we can show:
Since , then:
If is an eigenvector of with eigenvalue , such that , then is an eigenvector of with the same eigenvalue:
This shows that matrix similarity preserves the spectrum or set of eigenvalues, which is a fundamental characteristic of linear transformations.
