<commit_before>---
published: true
title: Dispaly Math equations and symbols
category: Latex
tags: 
  - latex
layout: post
---

\\[ \mathbf{X} = \mathbf{Z} \mathbf{P^\mathsf{T}} \\]

$$ \mathbf{X} \_{n,p} = \mathbf{A} \_{n,k} \mathbf{B} \_{k,p} $$

$$ \mathsf{Data = PCs} \times \mathsf{Loadings} $$

$$a^2 + b^2 = c^2$$

The question is that:

We have two matrices $A,B\in \mathbf{C}^{n\times n}$, A is nonsingular and B is singular, let $||\cdot ||$ be $\textbf{any}$ matrix norm, prove

$||A-B||\geq 1/||A^{-1}||$.

My idea is 

$||(A-B)||\cdot||A^{-1}||\geq||A^{-1}(A-B)||=||I-A^{-1}B||$

And I am confused here. How to calculate the right side?

$\color{red}{Note:}$ the original question may be wrong when the norm doesn't have sub-multiplicative property. See the comments below.

HINT:

$B= A+ (B-A) = A\cdot( I - A^{-1}(A-B) ) = A \cdot (I - C)$

where $C = A^{-1}(A-B)$.  Assume that $||A-B||< ||A^{-1}||^{-1}$. Then $||C||\le ||A^{-1} || \cdot ||A-B|| < 1$. Now use the following fact:

If $C$ is a matrix with norm $||C||<1$ then the series $\sum_{n\ge 0} C^n$ is absolutely convergent and its sum is the inverse of $I-C$. In particular: $||C||<1$ implies $(I-C)$ invertible. 

Use the above to conclude $B$ is invertible, contradiction.

<commit_msg>display mathjax<commit_after>---
published: true
title: Dispaly Math equations and symbols
category: Latex
tags: 
  - latex
layout: post
---

\\[ \mathbf{X} = \mathbf{Z} \mathbf{P^\mathsf{T}} \\]

\\[ \mathbf{X} \_{n,p} = \mathbf{A} \_{n,k} \mathbf{B} \_{k,p} \\]

$$ \mathsf{Data = PCs} \times \mathsf{Loadings} $$

$$a^2 + b^2 = c^2$$

\\[ J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma (m + \alpha + 1)} {\left({ \frac{x}{2} }\right)}^{2m + \alpha} \\]

$$ J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma (m + \alpha + 1)} {\left({ \frac{x}{2} }\right)}^{2m + \alpha} $$

$$ x^{y^z}=(1+{\rm e}^x)^{-2xy^w} $$

$$ \vec{a} \cdot \vec{b}=0 $$

$$ \alpha \beta　\gamma　\Gamma　\delta　\Delta　\epsilon \varepsilon　　\zeta　\eta　\theta　\Theta　\vartheta \iota　\kappa $$
$$　\lambda　\Lambda　\mu　　\nu　\xi　\Xi　　\pi　\Pi　\varpi　　\rho　\varrho　　\sigma　\Sigma　\varsigma　　\tau　$$
$$　\upsilon　\Upsilon  \phi　\Phi　\varphi　\chi　　\psi　\Psi　\omega　\Omega $$





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