### Eigenvectors Eigenvectors are the nonzero vectors $v$ such that $Av = \lambda v$, where $\lambda$ is the associated eigenvalue. Characteristic Polynomial $\det(A-\lambda I) = 0$ ### Singular Value Decomposition The Full SVD, $X = U\Sigma V^{*}$, and the economy SVD, $X = \hat{U}\hat{\Sigma}V^{*}$), can exactly represent any complex matrix $X$. The economy SVD reduces the size of the matrices when $n>m$ as illustrated below: ![[Pasted image 20260519171652.png|300]] The economy SVD: $ X = \begin{bmatrix} | & | & & |\\ x_{1} & x_{2} & \dots & x_{m} \\ | & | & & | \end{bmatrix} = \underbrace{ \begin{bmatrix} | & | & & |\\ u_{1} & u_{2} & \dots & u_{m} \\ | & | & & | \end{bmatrix} }_{ \text{Left Singular Vectors} } \underbrace{ \begin{bmatrix} \sigma_{1} & 0 & \dots \\ 0 & \sigma_{2} & 0 & \dots \\ \vdots & 0 & \ddots & \\ & & & \sigma_{m} \end{bmatrix} }_{ \text{Singular Values} } \underbrace{ \begin{bmatrix} — & v_{1} & — \\ — & v_{2} & — \\ & \vdots & \\ — & v_{m} & — \end{bmatrix} }_{ \text{Right Singular Vectors} } $ - $U$ and $V$ are unitary: $UU^{*} = U^{*}U =I$ and ${} VV^{*} = V^{*}V = I {}$ - $\{u_{1}, u_{2}, \dots u_{}\}$ is a orthonormal basis of $C(A)$ - $\{v_{1}, v_{2}, \dots v_{}\}$ is a orthonormal basis of $N(A)$ - Singular values are in decreasing order: $\sigma_{1}\geq \sigma_{2} \geq \dots\geq \sigma_{r} \geq 0$ Pseudoinverse $A^{\dagger} = \sum \frac{1}{\sigma_{i}}v_{i}u_{i}^{*}$ Projection matrix onto $C(A) = AA^{\dagger}$ Projection onto $N(A) = A^{\dagger}A$ ### Regression ### Diagonalization ### Rank-Nullity Theorem If $A$ is a $m$ x $n$ matrix, $\underbrace{ \text{rank}(A) }_{ \text{pivots} }+\underbrace{ \text{dim}N(A) }_{ \text{free variables} }=n$