📂Matrix Algebra

Matrix Similarity

Definition¹

A square matrix $A$, $B$ and an invertible matrix $P$ are said to be $B$ is similar to $A$ if the following equation holds.

$$ B = P^{-1} A P $$

Description

The reason why it is called similar is because similar matrices share many important properties. This is called similarity invariant or invariant under similarity.

Conjugate

When the given equation is expressed for $B$,

$$ B = P^{-1} A P $$

it is easy to see that the similarity relation is symmetric. Algebraically, it could be said that $A$ and $B$ are conjugate to each other with respect to $P$.

Theorem

Let’s assume $A$ and $B$ are similar matrices.

• Determinant: The determinant of $A$ and $B$ are the same.

• Invertibility: If $A$ is an invertible matrix, then $B$ is also an invertible matrix.

• Rank: The rank of $A$ and $B$ is the same.

• Nullity: The nullity of $A$ and $B$ is the same.

• Trace: The trace of $A$ and $B$ is the same.

• Characteristic Equation: The characteristic equation of $A$ and $B$ is the same. (Proof)

  • Eigenvalues: The eigenvalues of $A$ and $B$ are the same.

• Eigenvectors: If $\lambda$ is an eigenvalue of $A$, then the dimension of the eigenspace corresponding to $\lambda$ for $A$ and the dimension of the eigenspace corresponding to $\lambda$ for $B$ are the same.