Given two matrices $\mathbf{A}$, $\mathbf{B}$ that are known to be similar and invertible, is there a way (formula or algorithm) to determine $\mathbf{R}$ such that $\mathbf{B = R}\,\mathbf{A}\,\mathbf{R}^{\, - \,\mathbf{1}} $, without passing through the Jordan decomposition of $\mathbf{A}$ and $\mathbf{B}$ ?
2026-03-30 06:43:04.1774852984
Direct relation between similar matrices
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A practical way is to solve the linear matrix equation $\mathbf B\mathbf R - \mathbf R \mathbf A=0$ which is equivalent to the linear equation $(\mathbf I\otimes \mathbf B - \mathbf A^T\otimes\mathbf I)vec(\mathbf R) = 0$. The equation has many solutions and each generic solution satisfy $\det(\mathbf R)\ne 0$.