Problem A.25 - Matrix Harmony ⇢ Two Matrices, One Set of Eigenvectors: Intro to QM Appendix

⍟ 𝐀𝐛𝐨𝐮𝐭 𝐓𝐡𝐢𝐬 𝐕𝐢𝐝𝐞𝐨 ⍟
This video explores simultaneous diagonalization in linear algebra, focusing on two seemingly distinct matrices sharing the same eigenvectors. We examine Hermitian matrices A and B, demonstrating their normality and commutativity. Through characteristic equations and Gaussian elimination, we derive the eigen decomposition, revealing shared eigenvectors and distinct eigenvalues for both matrices.

• 𝙿𝚛𝚘𝚋𝚕𝚎𝚖 𝙱𝚛𝚎𝚊𝚔𝚍𝚘𝚠𝚗 𝚃𝚒𝚖𝚎 𝚂𝚝𝚊𝚖𝚙𝚜:
00:00 - Intro & Background.
00:12 - Problem Statement.
00:48 - Part (a): Commutativity.
04:46 - Part (b): Eigen Decomposition.
14:55 - Part (c): Simultaneously Diagonalizability.
19:00 - Concluding Remarks.
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☞📚📖📓= Griffiths, David J., and Darrell F. Schroeter. “Appendix: Linear Algebra.” 𝘐𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘵𝘪𝘰𝘯 𝘵𝘰 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘔𝘦𝘤𝘩𝘢𝘯𝘪𝘤𝘴, 3rd ed., Cambridge University Press, 2018, pp. 464–485.

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Science is a phenomenal exploration of nature. We hope to hone our skills of problem solving by exposing ourselves to multiple contexts. In doing so, it can sometimes be challenging to see the connection between topics. I yearn to understand 𝙝𝙤𝙬 these aspects of physics, unite together. To accomplish this, I'll cover all of my old textbooks through QFT; the convergence point of the many modern scientists! These posts are very much in a "𝘯𝘰𝘵𝘦𝘴 𝘵𝘰 𝘴𝘦𝘭𝘧" style. 𝙈𝙮 𝙝𝙤𝙥𝙚 is that by sharing this exploration, I can help others navigate the beautiful world of mathematics & physics through problems and examples, connecting the mathematical tools to their physical ramifications.

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