Eigenvalues of A Hermitian Matrix
Majorization
Majorization is a concept used to evaluate the degree of “equality” between two vectors. Definition 1.1 (Majorization): Given $$x,y\in\mathbb{R}^n$$, we say $$x$$ is majorized by $$y$$ if \(\begin{equation} \left\{ \begin{array}{cl} \sum_{i=1}^k x_i^{\downarrow}\leq\sum_{i=1}^k y_i^{\downarrow}\,, & \quad k=1,\ldots, n-1\,, \\ \sum_{i=1}^n x_i^{\downarrow}\leq\sum_{i=1}^n y_i^{\downarrow}\,, & \end{array} \right. \end{equation}\) where $$x_i^{\downarrow}$$ and $$y_i^{\downarrow}$$ represent the $$i$$-th elements of the non-increasing rearrangement of $$x$$ and $$y$$, respectively.
This blog introduces the [@Marshall2011Inequalities] This theme supports rendering beautiful math in inline and display modes using MathJax 3 engine. You just need to surround your math expression with $$, like $$ E = mc^2 $$. If you leave it inside a paragraph, it will produce an inline expression, just like \(E = mc^2\).
Diagonal Elements and Eigenvalues of a Hermitian Matrix
To use display mode, again surround your expression with $$ and place it as a separate paragraph. Here is an example:
Eigenvalues of a Hermitian Matrix
\(\sum_{k=1}^\infty |\langle x, e_k \rangle|^2 \leq \|x\|^2\)
You can also use \begin{equation}...\end{equation} instead of $$ for display mode math. MathJax will automatically number equations:
\begin{equation} \label{eq:cauchy-schwarz} \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \end{equation}
and by adding \label{...} inside the equation environment, we can now refer to the equation using \eqref.
Note that MathJax 3 is a major re-write of MathJax that brought a significant improvement to the loading and rendering speed, which is now on par with KaTeX.
References
- Horn, R. A., Johnson, C. R. (1990). Matrix Analysis. Cambridge University Press. ISBN: 0521386322
- Marshall, A. W., Olkin, I.,, Arnold, B. C. (2011). Inequalities: Theory of Majorization and its Applications (Vol. 143). Springer.