Similarity
Similarity
Let and be n x n matrices. is similar to if there exists an invertible n x n matrix such that . Obviously, this relation is symmetric i.e. is similar to implies is also similar to (since when ). Therefore, sometimes we just say and are similar.
As we know, the square matrix of a linear transformation means for any column vector , whose entries are (by default) coordinates using the standard coordinate system in . However, if we use another coordinate system defined by a new basis for and express the same linear transformation as a matrix i.e. , where now both and are column vectors whose entries are coordinates using the new coordinate system, it can be shown that , where is the n x n matrix formed by the column vectors of the new basis in the standard coordinate system. In other word, and are similar. P is usually called the change-of-coordinates matrix.
In the applet below, you can see the effect on the matrix representation of a linear transformation under the change of coordinates. First, you can define the linear transformation by setting the vectors in the "Range". Then you can set the two vectors that define the new coordinate system, which is represented by the dotted green gridlines (when are linearly independent). Then you can freely drag the vector and see how and are related using two different coordinate systems.
Properties of Similar Matrices
Suppose A and B are similar. The following theorem shows that they have a lot in common:
Theorem: Let and be n x n matrices such that and are similar. Then we have the following:
- They have the same determinant.
- They have the same characteristic equation and hence the same eigenvalues (with the same multiplicities).
- They have the same rank i.e.
Question
If and are two n x n matrices such that they have the same characteristic equation i.e. they have the same eigenvalues (with multiplicities), are they similar?