Matrix Diagonalisation and Change of Basis

Here’s the $\LaTeX$ code of my diagram for matrix diagonalisation to be used on Discord.

Why do matrix diagonalisation on square matrix $P$?

If we can find a diagonal matrix $D$ and a square matrix $Q$ such that $P = QDQ^{-1}$, then we can easily compute $(P + \lambda I)^n$ for any scalar $\lambda$ and integer $n$ because $D^n$ is easy to compute.

\[\begin{tikzcd}
    {{}} & {{}} & \cdots & {} \\
    {{}} & {{}} & \cdots & {{}}
    \arrow["P", from=1-1, to=1-2]
    \arrow["{Q^{-1}}"', from=1-1, to=2-1]
    \arrow["D"', from=2-1, to=2-2]
    \arrow["Q"', from=2-2, to=1-2]
    \arrow["P", from=1-2, to=1-3]
    \arrow["D"', from=2-2, to=2-3]
    \arrow["P", from=1-3, to=1-4]
    \arrow["D"', from=2-3, to=2-4]
    \arrow["Q"', from=2-4, to=1-4]
\end{tikzcd}\]

After viewing the power of the Discord bot $\TeX{}$it, which renders $\LaTeX$ code on Discord, I gave up spending more time on exploring more functionalities of $\KaTeX$ (say, commutative diagrams) because Discord and $\LaTeX$ spread math knowledge much better than a static blog for basic math: the former allows instant feedback from the reader. The later is better for taking notes. To display more complicated graphics, I can compile to PDF first, then use dvisvgm with -P for --pdf. (The small -p selects --page=ranges.)

On Discord, the above code block is to be typeset as plain $\LaTeX$ code.

• MathBot: =texp
• $\TeX{}$it: ,tex

To get the PDF then SVG, we need to get rid of the surrounding \[\] since we’re going to use the standalone mode. It would be nice to have a border of 2pt.

\documentclass[tikz, border=2pt]{standalone}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
...
\end{tikzcd}
\end{document}

The above diagram is rendered with

• -P: from PDF
• -o: specify output file name
• -T "S 2.5": transform by a scaling factor of 2.5.

Inside some arrow[...], there’s an extra '. In fact, that’s from quiver’s generated $\LaTeX$ code and that determines whether the arrow’s label text is above/below the arrow.

original quiver diagram

To finish, I’ll include another TikZ diagram illustrating the matrix representation of linear transformation with respect to two bases.

\[\begin{tikzcd}
    {[\cdot]_E} & {v_i} & {T(v_i)} \\
    {[\cdot]_B} & {e_i=[v_i]_B} & {[T(v_i)]_B}
    \arrow["{[T]_B}"', from=2-2, to=2-3]
    \arrow["T", from=1-2, to=1-3]
    \arrow["{?}", shift left=1, harpoon, from=1-2, to=2-2]
    \arrow["{B [v_i]_B}", shift left=1, harpoon, from=2-2, to=1-2]
    \arrow["{?}", shift left=1, harpoon, from=1-3, to=2-3]
    \arrow["{B[T(v_i)]_B}", shift left=1, harpoon, from=2-3, to=1-3]
\end{tikzcd}\]

original quiver diagram