Matrix similarity

• categories: Linear algebra, Definition

Definition
Two square matrices $A$ and $B$ of the same size are said to be similar if there exists an invertible matrix $P$ such that:
$B=P^{-1}AP.$
Intuition
Matrix similarity indicates that $A$ and $B$ represent the same linear transformation under different bases. The matrix $P$ performs the change of basis. Similar matrices share many fundamental properties, as their core behaviors (eigenvalues, determinant, trace, etc.) are preserved under similarity transformations.

Key Properties

1. Equivalence Relation:

   • Similarity is reflexive, symmetric, and transitive.

2. Preservation of Spectral Properties:

   • Similar matrices have the same eigenvalues (including algebraic and geometric multiplicities).
   • Similar matrices have the same characteristic polynomial.

3. Trace and Determinant:

   • $\text{tr}(A)=\text{tr}(B)$ and $\det (A)=\det (B)$ if $A\sim B$.

4. Rank:

   • Similar matrices have the same rank.

5. Canonical Forms:

   • Similar matrices can be reduced to simpler forms, such as the Jordan Normal Form or Diagonal Form (if diagonalizable).

6. Similarity vs. Congruence:

   • Similarity uses $P^{-1}AP$, while congruence uses $P^{T}AP$ or $P^{\ast }AP$ in the context of symmetric or Hermitian matrices.

Applications

1. Diagonalization:

   • A matrix $A$ is similar to a diagonal matrix $D$ if $A$ is diagonalizable. In this case, $A=PD{P}^{-1}$, where $D$ contains the eigenvalues of $A$.

2. Canonical Representations:

   • Similarity is used to reduce matrices to their Jordan Normal Form, simplifying computations in advanced linear algebra.

3. Change of Basis:

   • Similarity transformations describe how a matrix changes under a change of basis in a vector space.

4. Control Theory:

   • Similar matrices describe equivalent dynamical systems in different coordinate systems.

5. Quantum Mechanics:

   • Used to express operators in alternative bases while preserving eigenvalues (observable quantities).

Examples

1. Diagonalization:
   Let $A=\left[\begin{array}{cc}4& 1\\ 0& 2\end{array}\right]$ and $P=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$. Compute:
   $P^{-1}AP=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right],$
   showing that $A$ is similar to a diagonal matrix.

2. Non-Diagonalizable Case:
   Let $A=\left[\begin{array}{cc}2& 1\\ 0& 2\end{array}\right]$. $A$ is not diagonalizable, but it is similar to its Jordan Normal Form:
   $P^{-1}AP=\left[\begin{array}{cc}2& 1\\ 0& 2\end{array}\right],$
   where $P$ represents the change of basis.