Vectores
| \(vv^\top\) |
\( \mathcal{M}_{n\times n} \) (rango 1) |
Operador de Proyección Ortogonal en \(v\) \(P_v=\frac{v v^\top }{ v^\top v}\) |
| \(v^\top v\) |
\( \mathbb{R} \) |
\(\langle v, v \rangle = \|v\|^2\) |
| \(vu^\top\) |
\( \mathcal{M}_{n\times n} \) (rango 1) |
Descomposicion RC \(\chi_A(\lambda) = (-\lambda)^{n-1} (v^\top u - \lambda)\) |
| \(v^\top u\) |
\( \mathbb{R} \) |
\(\langle v, u \rangle = \|v\|\|u\|\cos\theta\) |
Matrices
| Stiefel |
\(Q \in \mathcal{M}_{m\times n} \)
\( Q^\top Q = \mathbb{I} \) |
- Isometría
- Columnas ortonormales
- \(Q Q^\top = P_Q\)
- Idempotente, Invariante en \(Q\)
|
| Ortogonal |
\(Q \in \mathcal{M}_{n\times n} \) |
- Caso particular de Stiefel, ademas cuadrada
- Autovectores \(\in \mathbb{C}^n\) ortogonales (\(v^*w=0\))
- \(Q^{-1}= Q^\top \)
- \(\lambda_i\in\mathbb{C},\ |\lambda_i| = 1\)
- Isometria (preserva normas y angulos)
- si \(\lambda\in \mathbb{R} \) entonces \(\lambda=\pm1\)
- Rotacion \(\det(Q)=1\), o Reflexion \(\det(Q)=-1\)
|
| Simétrica |
\(S \in \mathcal{M}(\mathbb{R})_{n\times n} \)
\(S = S^\top\) |
- Autovectores ortogonales
- \(\lambda \in \mathbb{R} \)
- \(S=Q\Lambda Q^\top\)
|
| Simétrica y Ortogonal |
\(S \in \mathcal{M}(\mathbb{R})_{n\times n} \)
\(S = S^\top\)
\(S^{-1}=S^\top\) |
|
| Hermítica |
\(S \in \mathcal{M}(\mathbb{C})_{n\times n} \)
\(S = S^* = \overline{S}^\top \) |
- Autovectores ortogonales (en \(\mathbb{C}\), \(v^*w=0\))
- \(\lambda \in \mathbb{C}\)
- \(S=Q\Lambda Q^\top \)
|
| Antisimétrica |
\(A\)
\( A^\top =-A\) |
- \(\lambda\) imaginario puro o cero
|
| Householder |
\(H\) |
- \(H = (I-2vv^\top)u\)
- Simetrica (proyección ortogonal)
- Involutiva (\(H^2= \mathbb{I} \))
- Ortogonal (isometría)
|
Proyecciónes y Reflexiónes
| Proyección de \(u\) sobre \(v\) |
\(u_v = (vv^\top)u = v\langle v, u \rangle\) |
| Proyección de \(u\) sobre \(v^\perp\) |
\(u_{v^\perp} = (I-vv^\top)u\) |
| Proyección de \(u\) sobre \(C(Q)\) |
\(QQ^\top u\) |
| Reflección de \(u\) sobre hiperplano \(v^\perp\) |
\(H = (I-2vv^\top)u\) |
| Idempotencia |
\(P^2=P\) |
| Invarianza |
\(v=Qx \implies Pv = v\) |
| Simetrica |
\(P= P^\top \) |
| Ortogonal |
\((Py)^\top(x-Px)=0, \forall y\) |
| Semidefinida Positiva |
\(\succeq 0\) |
Factorizaciones
| \(A = CR\) |
|
|
| \(A = X\Lambda X^{-1}\) |
\(A\in \mathcal{M}_{n\times n} \) con \(n\) Autovectores l.i. |
\(AX=X\Lambda\) |
| \(S = Q\Lambda Q^\top\) |
\(S=S^\top\), \(S\in \mathcal{M}_{n\times n} \) |
Teorema Espectral |
| \(A = U\Sigma V^\top\) |
|
|
Normas Vectoriales
| \(\ell_0\) |
\( \left\| v \right\| _0 = \#\{i: v_i \neq 0\}\) |
No es norma |
| \(\ell_1\) |
\( \left\| v \right\| _1 = \sum_{i=1}^{n} |v_i| = |v_1| + |v_2| + \dots + |v_n|\) |
Manhattan |
| \(\ell_2\) |
\( \left\| v \right\| _2 = \sigma_1\) |
Euclidea |
| \(\ell_p\) |
\( \left\| v \right\| _p = \left( \sum_{i=1}^{n} |v_i|^p \right)^{1/p}\) |
|
| \(\ell_\infty\) |
\( \left\| v \right\| _\infty = \max |v_i| \ \ i=1\dots n\) |
Máximo |
| \(S\) |
\( \left\| v \right\| _S = \sqrt{ v^\top Sv}\) |
S / Energía |
| Positividad |
\( \left\| v \right\| >=0, \quad \left\| v \right\| = 0 \implies v=0\) |
| Homogeneidad |
\( \left\| cv \right\| = |c| \cdot \left\| v \right\| \) |
| Triángular |
\( \left\| v+w \right\| \le \left\| v \right\| + \left\| w \right\| \) |
| Polarizacion |
\(\langle v, w \rangle = \frac{1}{4} \left( \left\| v + w \right\| ^2 - \left\| v - w \right\| ^2 \right)\) |
| Paralelogramo |
\( \left\| v+w \right\| ^2 + \left\| v-w \right\| ^2 = 2 \left\| v \right\| ^2 + 2 \left\| w \right\| ^2\) |
| Cauchy-Schwarz |
\(| v^\top w| \leq \left\| v \right\| _2 \left\| w \right\| _2\) |
Producto interno Hilbert-Schmidt
\[\langle A, B \rangle_{HS} = \text{tr}( A^\top B) = \sum_{j=1}^n\sum_{i=1}^ma_{ij}b_{ij}=vec(A) \cdot vec(B)\]
| Simetría |
\(\langle A, B \rangle=\langle B, A \rangle\) |
| Linealidad |
\(\langle cA, B \rangle=c\langle A, B \rangle\) |
| Positividad |
\(\langle A, A \rangle \geq 0, \ A \neq 0\) |
Normas Matriciales
| Operador / Espectral |
\(\displaystyle \left\| A \right\| _2 = \max_{x\neq0}{\frac{ \left\| Ax \right\| }{ \left\| x \right\| }} = \max_{ \left\| x \right\| =1}{ \left\| Ax \right\| }\) |
\(\displaystyle \sigma_1\) |
Norma \(\ell_2\) |
|
\(\displaystyle \left\| A \right\| _1=\max_{1 \leq j \leq n}{ \sum_{i=1}^m{ |a_{ij}| } }\) |
|
Norma \(\ell_1\) |
|
\(\displaystyle \left\| A \right\| _\infty=\max_{1 \leq i \leq m}{ \sum_{j=1}^n{ |a_{ij}| } }\) |
|
Norma \(\ell_\infty\) |
| Nuclear / Traza |
\(\displaystyle \left\| A \right\| _N = \sigma_1 + \dots + \sigma_r\) |
\(\displaystyle \sum_{i=1}^r{\sigma_i}\) |
|
| Frobenius |
\(\displaystyle \left\| A \right\| _F = \left( \sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^2 \right)^{ ½ }\) |
\(\displaystyle \left(\sum{\sigma_i^2}\right)^ ½ \) |
Hilbert-Schmidt |
| \( \left\| A \right\| _2 = \sigma_1\) |
|
| \( \left\| A \right\| _F = (\sigma_1^2 + \dots + \sigma_r^2)^ ½ = \sqrt{\langle A, A \rangle_{HS}} = \sqrt{ \text{traza}( A^\top A) }\) |
|
| \( \left\| A \right\| _N = \sum_{i=1\dots r}{\sigma_i}\) |
|
| \( \left\| AB \right\| \leq \left\| A \right\| \left\| B \right\| \) |
|
| \( \left\| QA \right\| = \left\| AQ \right\| = \left\| A \right\| \) |
Invarianza Ortogonal (Frobenius, Operador, Nuclear) |
| \( \left\| A \right\| _2\leq \left\| A \right\| _F\leq \left\| A \right\| _N\) |
|
SVD
\[A = U\Sigma V^\top \] \[A^+ = ( A^\top A)^{-1} A^\top \]