I've come across an nxn matrix whose entry in row i and col j is min(i,j). For example, the 4x4 version would be

\[

\left(

\begin{array}{cccc}

1 & 1 & 1 & 1\\

1 & 2 & 2 & 2\\

1 & 2 & 3 & 3\\

1 & 2 & 3 & 4

\end{array}

\right)

\]

Does anybody recognize this class of matrices? Does it have a name? I'd like to research what is known about it.

... also does anyone know how possible it is to get MathJax or similar working in an android app. I'm thinking of forking https://github.com/Vavassor/Tusky and attempting to get \(\mathrm{\LaTeX}\) to work in it...

A student pointed out to me yesterday that the values of \(\sin\) at 0, \(\tfrac\pi6\), \(\tfrac\pi4\), \(\tfrac\pi3\), \(\tfrac\pi4\) are \(\frac{\sqrt0}2\), \(\frac{\sqrt1}2\), \(\frac{\sqrt2}2\), \(\frac{\sqrt3}2\) and \(\frac{\sqrt4}2\). Amazed I'd never been told this way to remember them before.

#proofinatoot that Chebyshev polynomials, defined by \(T_n(x)=\cos(n\cos^{-1}x)\) are given by \[T_0(x)=1,\quad T_1(x)=x,\quad T_{k+1}(x)=2xT_k(x)-T_{k-1}(x)\]

\begin{align*}T_0(x)&=\cos(0\cos^{-1}x)\\&=1.\end{align*}

\begin{align*}T_1(x)&=\cos(1\cos^{-1}x)\\&=x.\end{align*}

For \(n>=2\), use \(\cos(A)+\cos(B)=2\cos(\tfrac{A+B}2)\cos(\tfrac{A-B}2)\): \begin{align*}T_{n+1}(x)+T_{n-1}(x)&=\cos((n+1)\cos^{-1}x)+\cos((n+1)\cos^{-1}x)\\

&=2\cos(n\cos^{-1}(x))\cos(\cos{-1}x)\\

&=2xT_n(x)

\end{align*}

— I like maths —

http://mscroggs.co.uk

Joined May 2017