Definition of similarity transform

Definition: A similarity transform (or similarity transformation) is the operation of conjugating a matrix $A$ by an invertible matrix $P$:

$$A \mapsto P^{-1}AP$$

The matrix $P$ is called the transformation matrix or change-of-basis matrix.

Properties preserved under similarity transform:

• Eigenvalues
• Determinant
• Trace
• Rank
• Characteristic polynomial
• Minimal polynomial

Geometric meaning: If $A$ represents a linear operator in one coordinate system, then $P^{-1}AP$ represents the same operator in a different coordinate system defined by the columns of $P$.

Common applications:

• Diagonalization: $P^{-1}AP = D$ (when $A$ is diagonalizable)
• Jordan form: $P^{-1}AP = J$
• Schur form: $Q^*AQ = T$ (unitary similarity)