Summary & lab

Linear algebra lab

The interactive demo below illustrates the effects of a transformation matrix by applying the transformation on a two-dimensional grid. It also shows the eigen vectors.

Summary linear algebra

\[ 👉\ \text{Matrix}:\ A= \begin{pmatrix} a&b\\ b&d \end{pmatrix} \]

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\[ 👉\ \text{Eigen values/eigen vectors}:\ Ae_i=\lambda_i e_i \]

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\[ 👉\ \text{Spectral theorem}:\ e_1\perp e_2 \]

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\[ 👉\ \text{Orthonormal matrix}:\ Q=(e_1\ e_2) \]

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\[ 👉\ \text{Diagonale matrix}:\ \Lambda= \begin{pmatrix} \lambda_1&0\\ 0&\lambda_2 \end{pmatrix} \]

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\[ 👉\ \text{Spectral decomposition}:\ \boxed{A=Q\Lambda Q^T} \]