# 3M1 Mathematical Methods Tripos Revision

Posted by Jingbiao on April 10, 2021

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## Linear Algebra

#### Hermitian

• Conjugate Transpose: $$M^H = \overline{M^T} = \overline{M}^T$$
• Hermitian: if $M^H = M$
• $(AB)^H = B^H A^H$
• $A^H A$ must be hermitian
• Unitary matrix: if $M^H = M^{-1}$

• Hermitian positive definite Matrix: $$x^H M x > 0 \forall x \in \mathcal{C}^n\backslash 0$$

#### Vector/Matrix Norms

##### Vector Norm
• Vector $l_p$ norms: $$\norm{x} = (\sum_{i} |x_i|^p )^{1/p}$$

• Infinite norm just find the maximum term of the vector $$\norm{x}_{\infty} = \max_i |x_i|$$ which is also knownas the maiximum norm.
• Matrix induced norm: $$\norm{x}^2_{A} = x^H A x$$

• Properties:

• Linearity: $$\norm{kx} = |k| \norm{x}$$
• Triangle inequality: $$\norm{x+y} \le \norm{x} + \norm{y}$$
• Another inequality: $$\norm{xy} \le \norm{x}\norm{y}$$
##### Matrix Norm
• Operator norms $$\norm{A} = \max_{\forall x \in \mathcal{C}^n\backslash 0} \frac{\norm{Ax}}{\norm{x}}$$

• This norm measures the maximum amount by which the matrix $A$ can re-scale a vector $x$
• 1-norm: $$\norm{A} =\max_j \sum_{i=1}^n |a_{ij}|$$ which is column of A with maximum $l_1$ norm
• $\infty$ norm: $$\norm{A} = \max_i \sum_{j}^n|a_{ij}|$$ which is row of A with maximum $l_1$ norm

• $l_2$ norm: $$\norm{A} = \sqrt{\lambda_{\max}(A^H A)}$$
##### Condition number
• $\kappa(A) = \norm{A}\norm{A^{-1}}$
• For the 2-norm: $$\kappa_2(A) = \frac{\sqrt{\lambda_{\max}A^H A}}{\sqrt{\lambda_{\min}A^H A}}$$
• which is the max singular value over min singular value
• Since eigenvalue is reciprocal for matrix inverse, if A is Hermitian, then: $$\kappa_2(A) = \frac{|\lambda(A)|max}{|\lambda(A)|min}$$
• A matrix with a large condition number is ill-conditioned, which lead to instable computation $\rightarrow$ small error lead to large computation error

## Optimisation

• Linear Programming solved by using Simplex Algorithm

## Monte Carlo 