Problem A.24- Eigenvector Invariance in Simultaneously Diagonalizable Matrices: Intro to QM Appendix

⍟ 𝐀𝐛𝐨𝐮𝐭 𝐓𝐡𝐢𝐬 𝐕𝐢𝐝𝐞𝐨 ⍟
𝙎𝙞𝙢𝙪𝙡𝙩𝙖𝙣𝙚𝙤𝙪𝙨𝙡𝙮 𝘿𝙞𝙖𝙜𝙤𝙣𝙖𝙡𝙞𝙯𝙖𝙗𝙡𝙚 𝙈𝙖𝙩𝙧𝙞𝙘𝙚𝙨:
Matrices A and B are said to be 𝐬𝐢𝐦𝐮𝐥𝐭𝐚𝐧𝐞𝐨𝐮𝐬𝐥𝐲 𝐝𝐢𝐚𝐠𝐨𝐧𝐚𝐥𝐢𝐳𝐚𝐛𝐥𝐞 if there exists an invertible matrix P such that both P^−1AP and P^−1BP are diagonal matrices. This implies that:
• There must exist a common basis of eigenvectors for both matrices A and B.
• The matrices must commute, i.e., AB=BA.
𝘿𝙚𝙜𝙚𝙣𝙚𝙧𝙖𝙩𝙚 𝙀𝙞𝙜𝙚𝙣𝙫𝙖𝙡𝙪𝙚𝙨:
An eigenvalue is 𝐝𝐞𝐠𝐞𝐧𝐞𝐫𝐚𝐭𝐞 if it has more than one linearly independent eigenvector associated with it. This means:
• For an eigenvalue λ, there are multiple eigenvectors, and the dimension of the eigenspace (the geometric multiplicity) can be greater than one.
𝙄𝙢𝙥𝙡𝙞𝙘𝙖𝙩𝙞𝙤𝙣𝙨 𝙛𝙤𝙧 𝙎𝙞𝙢𝙪𝙡𝙩𝙖𝙣𝙚𝙤𝙪𝙨 𝘿𝙞𝙖𝙜𝙤𝙣𝙖𝙡𝙞𝙯𝙖𝙩𝙞𝙤𝙣 𝙬𝙞𝙩𝙝 𝘿𝙚𝙜𝙚𝙣𝙚𝙧𝙖𝙩𝙚 𝙀𝙞𝙜𝙚𝙣𝙫𝙖𝙡𝙪𝙚𝙨:
• 𝐂𝐨𝐦𝐩𝐥𝐞𝐱𝐢𝐭𝐲: Degeneracy adds complexity because it requires ensuring that the different eigenvectors corresponding to the same eigenvalue are consistent across both matrices.

• 𝙿𝚛𝚘𝚋𝚕𝚎𝚖 𝙱𝚛𝚎𝚊𝚔𝚍𝚘𝚠𝚗 𝚃𝚒𝚖𝚎 𝚂𝚝𝚊𝚖𝚙𝚜:
00:00 - Intro & Background.
00:11 - Problem Statement.
02:38 - Proof.
07:50 - Concluding Remarks.
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⍟ 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.

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