From 10b83af54da036b9f3122b5e82b18816c5acf386 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Thu, 12 Mar 2020 20:05:58 -0400 Subject: [PATCH] mjo-linear_algebra.tex: add \diag{} to build diagonal matrices. --- examples.tex | 12 +++++++----- mjo-linear_algebra.tex | 5 +++++ 2 files changed, 12 insertions(+), 5 deletions(-) diff --git a/examples.tex b/examples.tex index 1d79079..c3e22e7 100644 --- a/examples.tex +++ b/examples.tex @@ -176,14 +176,16 @@ matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the - set of its eigenvalues---is $\spectrum{L}$. Another matrix-specific - concept is the Moore-Penrose pseudoinverse of $L$, denoted by - $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is - $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$ + set of its eigenvalues---is $\spectrum{L}$. Another + matrix-specific concept is the Moore-Penrose pseudoinverse of $L$, + denoted by $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ + is $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$ real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$ respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically simplify because the ``$n$'' does not indicate the arity of a - Cartesian product in this case. + Cartesian product in this case. A handy way to represent the + matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on + the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$. The span of a set $X$ is $\spanof{X}$, and its codimension is $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex index d493b9c..c57e9b9 100644 --- a/mjo-linear_algebra.tex +++ b/mjo-linear_algebra.tex @@ -51,6 +51,11 @@ % The trace of an operator. \newcommand*{\trace}[1]{ \operatorname{trace}\of{{#1}} } +% The diagonal matrix whose only nonzero entries are on the diagonal +% and are given by our argument. The argument should therefore be a +% vector or tuple of entries, by convention going from the top-left to +% the bottom-right of the matrix. +\newcommand*{\diag}[1]{\operatorname{diag}\of{{#1}}} % The "rank" of its argument, which is context-dependent. It can mean % any or all of, -- 2.44.2