Similarity transformation

A similarity transformation sends a square matrix $A$ to $SA{S}^{-1}$ for an invertible matrix $S$. It changes the basis of a linear operator and preserves its characteristic polynomial and eigenvalues.

If a real matrix $B$ has a positive diagonal matrix $W$ such that $WB$ is a symmetric matrix, then $H=W^{1/2}B{W}^{-1/2}$ is a symmetric matrix related to $B$ by a similarity transformation: $B^{T}W=WB$ gives $H^T=H$. Thus $B$ is a diagonalizable matrix with real eigenvalues. If ${\mathbf{z}}^{T}WB\mathbf{z}\le 0$ for all real vectors $\mathbf{z}$, its eigenvalues are nonpositive. This converts symmetry in a weighted inner product into Euclidean symmetry.

Every unital algebra automorphism $F:M_n(\mathbb{C})\to M_n(\mathbb{C})$ has the form $F(X)=SX{S}^{-1}$ for some invertible $S$. One proof transports the matrix units through $F$, chooses compatible bases for their rank-one images, and reconstructs the common similarity transformation.