ASKSAGE: Sage Q&A Forum - Individual question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 17 Sep 2019 00:54:25 -0500Computation of a simultaneous block-diagonalizationhttps://ask.sagemath.org/question/47916/computation-of-a-simultaneous-block-diagonalization/Let $n$ be a positive integer and consider of finite set $S \subset M_n(\mathbb{C})$ such that $S^* = S$ (i.e. if $M \in S$ then $M^* \in S$). The algebra generated by $S$ is a finite dimensional $*$-algebra over $\mathbb{C}$, so is isomorphic to a direct sum of matrix algebras, i.e. there are $n_1 \le n_2 \le \dots \le n_r$ such that:
$$\langle S \rangle \simeq \bigoplus_{i=1}^r M_{n_i}(\mathbb{C})$$
I know how to get $\langle S \rangle$ by using `FiniteDimensionalAlgebra(CC,[M for M in S])`.
**Question**: How to compute with SageMath the change of basis $P$ such that for all $M \in S$, we have $P^{-1}MP$ block-diagonal as for the above decomposition?
*Remark*: When the matrices commute over each other, it is called a *simultaneous diagonalization*, and I know how to compute it using `jordan_form(transformation=True)` several times.
In some sense, what I am looking for in general is how to compute a *simultaneous block-diagonalization*Sat, 14 Sep 2019 10:33:58 -0500https://ask.sagemath.org/question/47916/computation-of-a-simultaneous-block-diagonalization/Comment by Sébastien Palcoux for <p>Let $n$ be a positive integer and consider of finite set $S \subset M_n(\mathbb{C})$ such that $S^* = S$ (i.e. if $M \in S$ then $M^* \in S$). The algebra generated by $S$ is a finite dimensional $*$-algebra over $\mathbb{C}$, so is isomorphic to a direct sum of matrix algebras, i.e. there are $n_1 \le n_2 \le \dots \le n_r$ such that: <br>
$$\langle S \rangle \simeq \bigoplus_{i=1}^r M_{n_i}(\mathbb{C})$$
I know how to get $\langle S \rangle$ by using <code>FiniteDimensionalAlgebra(CC,[M for M in S])</code>.</p>
<p><strong>Question</strong>: How to compute with SageMath the change of basis $P$ such that for all $M \in S$, we have $P^{-1}MP$ block-diagonal as for the above decomposition?</p>
<p><em>Remark</em>: When the matrices commute over each other, it is called a <em>simultaneous diagonalization</em>, and I know how to compute it using <code>jordan_form(transformation=True)</code> several times. <br>
In some sense, what I am looking for in general is how to compute a <em>simultaneous block-diagonalization</em></p>
https://ask.sagemath.org/question/47916/computation-of-a-simultaneous-block-diagonalization/?comment=47945#post-id-47945As this question seems to be not "really" about SageMath, I just posted it on MathOverflow: https://mathoverflow.net/q/341793/34538Tue, 17 Sep 2019 00:54:25 -0500https://ask.sagemath.org/question/47916/computation-of-a-simultaneous-block-diagonalization/?comment=47945#post-id-47945