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Eigenvalues & Eigenvectors

📝 𝗣𝗥𝗜𝗡𝗧𝗔𝗕𝗟𝗘 𝗡𝗢𝗧𝗘𝗦 https://drive.google.com/file/d/1BEDl... 𝗤𝗨𝗘𝗦𝗧𝗜𝗢𝗡𝗦 2:11 Example 1, 2 × 2 matrix. 8:52 Example 2, 2 × 2 matrix. 11:05 Example 3, 3 × 3 matrix. 16:33 Example 4, 3 × 3 matrix. 𝗟𝗘𝗔𝗥𝗡𝗜𝗡𝗚 𝗢𝗕𝗝𝗘𝗖𝗧𝗜𝗩𝗘𝗦 ♥ Find eigenvalues and eigenvectors when given a matrix. ♥ Find the basis vectors of an eigenspace of a given matrix. ♥ State the characteristic equation of a matrix. ♥ Determine the algebraic multiplicity and geometric multiplicity of an eigenvalue. 𝗦𝗨𝗠𝗠𝗔𝗥𝗬 𝗘𝗶𝗴𝗲𝗻𝘃𝗮𝗹𝘂𝗲𝘀 (scalars, λ) and 𝗲𝗶𝗴𝗲𝗻𝘃𝗲𝗰𝘁𝗼𝗿𝘀 (column vectors, 𝘃) satisfy the equation A𝘃 = λ𝘃. Note that A is a square matrix. The relationship A𝘃 = λ𝘃 means that, instead of doing annoying matrix multiplication A𝘃, we can simply do scalar multiplication λ𝘃. However, it is not always when we can substitute matrix multiplication with scalar multiplication. The goal of finding eigenvalues and eigenvectors is to find situations where A𝘃 = λ𝘃 occur. Find eigenvalues by solving det(A − λI) = 0, or equivalently, det(λI − A) = 0. Subtract λ from the main diagonal entries of A, find the determinant, set the determinant equal to zero, and solve for the values of λ. The polynomial in λ set equal to zero is called the 𝗰𝗵𝗮𝗿𝗮𝗰𝘁𝗲𝗿𝗶𝘀𝘁𝗶𝗰 𝗲𝗾𝘂𝗮𝘁𝗶𝗼𝗻. Find eigenvectors by solving (A − λI)𝘅 = 𝟬, or equivalently, (λI − A)𝘅 = 𝟬. The system has an infinite number of solutions. The vectors of the solution set are the eigenvectors. Another way a question can ask for eigenvalues/eigenvectors is to ask for the bases of the eigenspaces of a matrix. Each eigenspace is defined by an eigenvalue. The eigenvectors of an eigenvalue are the bases of that eigenspace. 𝗔𝗹𝗴𝗲𝗯𝗿𝗮𝗶𝗰 𝗺𝘂𝗹𝘁𝗶𝗽𝗹𝗶𝗰𝗶𝘁𝘆 means the number of instances an eigenvalue comes up as the root of the characteristic equation. 𝗚𝗲𝗼𝗺𝗲𝘁𝗿𝗶𝗰 𝗺𝘂𝗹𝘁𝗶𝗽𝗹𝗶𝗰𝗶𝘁𝘆 means the number of eigenvectors associated with a certain eigenvalue. Original content © 2025 Jung-Lynn Jonathan Yang, CC-BY-NC-ND ♥