Intuitions About Eigenvalues and Eigenvectors – A Visual Guide 🎥

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3:00A nonzero vector v is an eigenvector when $Av=\lambda v$. The scalar may be negative or zero, so the transformed vector can reverse direction or collapse to zero.

6:00If eigenvectors form a basis and $x=a_1v_1+a_2v_2$, linearity gives $Ax=a_1\lambda_1v_1+a_2\lambda_2v_2$. Each eigenvector component is scaled separately.

13:30With an invertible eigenvector matrix T, write $A=TDT^{-1}$. Then $A^k=TD^kT^{-1}$; a complete eigenbasis is required for this method.

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