Similarity Transformation

Let $$T$$ be an orthogonal matrix. Then, the matrix $$T^{-1}AT$$ is called similarity transformation. The matrices $$A$$ and $$T^{-1}AT$$ are called similar matrices.

What's the point of similarity transformation?
Similarity transformation is particularly useful if the resulting matrix $$T^{-1}AT$$ is diagonal or triangular etc. Then, computing eigenvalues, determinant, trace etc. become much more simpler. See Diagonalization. Also, similar matrices have the same eigenvalues, which is a critical property that leads to a real-life algorithm for eigenvalue computation (see QR iteration algorithm in Example 7.3.9 of ).