Two $n\times n$ matrices $A,B$ are similar if there exists some non-singular matrix $P$ such that $A=PBP^{-1}$. Do NOT use this tag when referring to similarity between matrices based on distance or another norm. Use this tag when the question involves similarity between matrices, or conjugacy in the General Linear Group of invertible matrices.

For some field $F$, matrices $A,B\in M_n(F)$ are similar if there exists some $P\in GL_n(F)$ such that $$A=PBP^{-1}$$ Similar matrices represent the same linear operator under two (possibly) different bases, with $P$ being the change of basis matrix.

Given $A,B\in GL_n(F)$, similarity is equivalent to conjugacy, and the conjugation map $c_P:A\mapsto PAP^{-1}$ is also known as a similarity transformation.

Similarity is a useful concept as it is an equivalence relation on $M_n(F)$ that preserves many key invariants, such as

- Rank
- Characteristic polynomial, and attributes that can be derived from it:
- Determinant
- Trace
- Eigenvalues, and their algebraic multiplicities

- Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used).
- Minimal polynomial
- Frobenius normal form
- Jordan normal form, up to a permutation of the Jordan blocks
- Index of nilpotence
- Elementary divisors, which form a complete set of invariants for similarity of matrices over a principal ideal domain

Because of this, it is often useful to find a similar matrix that is "simpler" than the matrix of study to analyze. Over any algebraically closed field $F$, every matrix is similar to a matrix in Jordan form.

Use this tag if your question involves computing, conjugating, normalizing, or simply using similar matrices.