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)
**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)