If (A) has (n) independent eigenvectors, form (S = [v_1 \dots v_n]). Then: [ A = S\Lambda S^-1 ] where (\Lambda = \textdiag(\lambda_1, \dots, \lambda_n)).
A=XΛX-1cap A equals cap X cap lambda cap X to the negative 1 power Λcap lambda lecture notes for linear algebra gilbert strang
While not written notes, the comment sections under Strang’s lectures (on the MIT OCW YouTube channel) often contain timestamps and summaries. Some dedicated viewers have created “companion notes” linked in the video descriptions. If (A) has (n) independent eigenvectors, form (S
The row picture focuses on the equations individually. Each linear equation represents a line (in 2D space), a plane (in 3D space), or a hyperplane (in higher dimensions). The solution to the system is the single intersection point where all these hyperplanes meet. The Column Picture The solution to the system is the single
orthogonal matrix containing the left singular vectors (eigenvectors of AATcap A cap A to the cap T-th power Σcap sigma : An diagonal matrix containing the singular values VTcap V to the cap T-th power : An
+-------------------+-------------------+ | Primal Space | Dual Space | | (in R^n) | (in R^m) | +---------------+-------------------+-------------------+ | | Row Space | Column Space | | Main Space | C(A^T) | C(A) | | | Dimension: r | Dimension: r | +---------------+-------------------+-------------------+ | | Nullspace | Left Nullspace | | Nullspace | N(A) | N(A^T) | | | Dimension: n-r | Dimension: m-r | +---------------+-------------------+-------------------+ The Four Fundamental Subspaces Column Space,