Problem A.28 - Similarity Invariants ⇢ Hidden Beauty of Matrix Diagonalization: Intro to QM Appendix

⍟ 𝐀𝐛𝐨𝐮𝐭 𝐓𝐡𝐢𝐬 𝐕𝐢𝐝𝐞𝐨 ⍟
Embark on a captivating journey through linear algebra as we explore the intricacies of matrix diagonalization and the remarkable invariance of trace and determinant under similarity transformations. This video looks into a compelling 2x2 Hermitian matrix problem, guiding you through each step of the diagonalization process. You'll witness the derivation of eigenvalues using the characteristic equation, followed by the determination and normalization of eigenvectors. The heart of our exploration lies in constructing the diagonalization transformation, where we'll transform our original matrix into its diagonal form. As we progress, we'll demonstrate the fascinating invariance of trace and determinant. This mathematical odyssey not only showcases the elegance of linear algebra but also highlights its fundamental role in quantum mechanics and beyond. Join us as we unravel these hidden mathematical connections and discover the beauty inherent in matrix transformations.

• 𝙿𝚛𝚘𝚋𝚕𝚎𝚖 𝙱𝚛𝚎𝚊𝚔𝚍𝚘𝚠𝚗 𝚃𝚒𝚖𝚎 𝚂𝚝𝚊𝚖𝚙𝚜:
00:00 - Intro & Background.
00:11 - Problem Statement.
01:02 - Part (a): Hermiticity.
01:46 - Part (b): Eigenvalues.
02:46 - Part (c): Eigenvectors.
07:33 - Part (d): Diagonalization.
09:24 - Part (e): Invariant Properties.
12:42 - Concluding Remarks.
----------------------------------------------------
⍟ 𝐒𝐮𝐩𝐩𝐨𝐫𝐭 𝐓𝐡𝐢𝐬 𝐂𝐡𝐚𝐧𝐧𝐞𝐥 ⍟
• ▶️ 𝘚𝘶𝘣𝘴𝘤𝘳𝘪𝘣𝘦 ▶️ ➜ http://tinyurl.com/4kd8wahb
• 🔎 𝘗𝘢𝘵𝘳𝘦𝘰𝘯 🔍 ➜   / curiousaboutscience  
• ☕ Buy Me a Coffee ☕ ➜ https://buymeacoffee.com/curiousabout...
----------------------------------------------------
⍟ 𝐂𝐫𝐞𝐝𝐢𝐭𝐬/𝐑𝐞𝐬𝐨𝐮𝐫𝐜𝐞𝐬 ⍟
☞📚📖📓= Griffiths, David J., and Darrell F. Schroeter. “Appendix: Linear Algebra.” 𝘐𝘯𝘵𝘳𝘰𝘥𝘶𝘤𝘵𝘪𝘰𝘯 𝘵𝘰 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘔𝘦𝘤𝘩𝘢𝘯𝘪𝘤𝘴, 3rd ed., Cambridge University Press, 2018, pp. 464–485.

•
----------------------------------------------------
⍟ 𝐌𝐢𝐬𝐬𝐢𝐨𝐧 ⍟
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.

#Curiousaboutscience

• Stay Curious & Happy Learning!

⇢ Share knowledge - tag a friend!
⇢ Subscribe for more!
⇢ Don't forget to turn on video notifications!
----------------------------------------------------