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# Definition of similarity transform

Put content here.**Definition:** A **similarity transform** (or **similarity transformation**) is the operation of conjugating a matrix $A$ by an invertible matrix $P$:
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$$A \mapsto P^{-1}AP$$
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The matrix $P$ is called the **transformation matrix** or **change-of-basis matrix**.
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**Properties preserved under similarity transform:**
- Eigenvalues
- Determinant
- Trace
- Rank
- Characteristic polynomial
- Minimal polynomial
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**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 $P$.
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**Common applications:**
- Diagonalization: $P^{-1}AP = D$ (when $A$ is diagonalizable)
- Jordan form: $P^{-1}AP = J$
- Schur form: $Q^*AQ = T$ (unitary similarity)

# Parents

* Similarity of matrices