#LyX 2.3 created this file. For more info see http://www.lyx.org/ \lyxformat 544 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass book \use_default_options true \maintain_unincluded_children false \language spanish \language_package default \inputencoding auto \fontencoding global \font_roman "default" "default" \font_sans "default" "default" \font_typewriter "default" "default" \font_math "auto" "auto" \font_default_family default \use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures true \graphics default \default_output_format default \output_sync 0 \bibtex_command default \index_command default \paperfontsize default \spacing single \use_hyperref false \papersize default \use_geometry false \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 1 \use_minted 0 \index Index \shortcut idx \color #008000 \end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style french \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{sloppypar} \end_layout \end_inset En Octave, todos los valores son matrices. Los números (con sintaxis \family typewriter [-+]?(( \backslash d+ \backslash .?| \backslash d* \backslash . \backslash d+)([eE][-+]? \backslash d+)?|[Ii]nf) \family default o \family typewriter ( \family default {número} \family typewriter \backslash +)? \family default {número} \family typewriter ?i \family default ) representan matrices \begin_inset Formula $1\times1$ \end_inset de números de doble precisión, y las cadenas de caracteres (con sintaxis \family typewriter '([^']|'')*' \family default o \family typewriter "([^ \backslash \backslash ']| \backslash \backslash \family default {escape} \family typewriter )*" \family default ) representan matrices fila de caracteres. \begin_inset ERT status open \begin_layout Plain Layout \backslash end{sloppypar} \end_layout \end_inset \end_layout \begin_layout Standard La expresión \family typewriter [ \begin_inset Formula $a_{1}$ \end_inset , \family default ... \family typewriter , \begin_inset Formula $a_{p}$ \end_inset ] \family default concatena horizontalmente las matrices \begin_inset Formula $a_{1}\in{\cal M}_{m\times n_{1}}(S)$ \end_inset hasta \begin_inset Formula $a_{p}\in{\cal M}_{m\times n_{p}}(S)$ \end_inset en una matriz en \begin_inset Formula ${\cal M}_{m\times\sum_{k=1}^{p}n_{k}}(S)$ \end_inset , y la sintaxis \family typewriter [ \begin_inset Formula $a_{11}$ \end_inset , \family default ... \family typewriter , \begin_inset Formula $a_{1p_{1}}$ \end_inset ; \family default ... \family typewriter ; \begin_inset Formula $a_{q1}$ \end_inset , \family default ... \family typewriter , \begin_inset Formula $a_{qp_{q}}$ \end_inset ] \family default hace esto en cada parte, resultando en \begin_inset Formula $q$ \end_inset matrices \begin_inset Formula $b_{k}\in{\cal M}_{m_{k}\times n}(S)$ \end_inset , y las concatena verticalmente en una \begin_inset Formula ${\cal M}_{\sum_{k=1}^{q}m_{k}\times n}(S)$ \end_inset . \end_layout \begin_layout Standard El operador \family typewriter + \family default suma matrices numéricas de igual tamaño y \family typewriter * \family default multiplica matrices o una matriz por un escalar. Llamamos vector a una matriz fila. Entonces \family typewriter \emph on a \emph default : \emph on b \family default \emph default genera el vector \begin_inset Formula $(a,a+1,\dots,b)$ \end_inset y \family typewriter \emph on a \emph default : \emph on t \emph default : \emph on b \family default \emph default genera el vector \begin_inset Formula $(a,a+t,\dots,b)$ \end_inset . Cuando es posible, \family typewriter \begin_inset Formula $A$ \end_inset \backslash \begin_inset Formula $B$ \end_inset \family default devuelve una matriz \begin_inset Formula $X$ \end_inset tal que \begin_inset Formula $AX=B$ \end_inset . \family typewriter \begin_inset Formula $A$ \end_inset ' \family default devuelve \begin_inset Formula $A^{*}$ \end_inset . \end_layout \begin_layout Standard \family typewriter \emph on A \emph default ( \emph on x \emph default , \emph on y \emph default ) \family default devuelve la submatriz de \family typewriter \emph on A \family default \emph default formada por las columnas con índice en el vector \family typewriter \emph on x \family default \emph default y las filas con índice en el vector \family typewriter \emph on y \family default \emph default , y \family typewriter \emph on A \emph default ( \emph on x \emph default ) \family default convierte la matriz en un vector concatenando las traspuestas de sus columnas y toma los elementos del vector con índice en el vector \family typewriter \emph on x \family default \emph default . Ambos vectores se pueden sustituir por \family typewriter : \family default para tomar todas las filas o columnas, y los índices empiezan por 1. \end_layout \begin_layout Standard Las expresiones son sentencias, y estas deben terminar por salto de línea si se quiere que se imprima su resultado o por \family typewriter ; \family default , seguido opcionalmente de salto de línea, si no. La sentencia \family typewriter \emph on A \emph default = \emph on expr \family default \emph default asigna a la variable \family typewriter \emph on A \family default \emph default el valor \family typewriter \emph on expr \family default \emph default , y \family typewriter \emph on A \emph default ( \emph on x \emph default , \emph on y \emph default ) = \emph on expr \family default \emph default o \family typewriter \emph on A \emph default ( \emph on x \emph default ) = \emph on expr \family default \emph default asigna los elementos de la submatriz a la izquierda del \family typewriter = \family default a los de la devuelta por la expresión, que debe ser del mismo tamaño. Si la variable no existe, se crea, y si la submatriz indicada supone que \family typewriter \emph on A \family default \emph default es más grande, esta se amplía y se rellena con ceros. \end_layout \begin_layout Section Funciones sobre matrices \end_layout \begin_layout Description \family typewriter cond( \series bold \emph on A \series default \emph default , \emph on p \emph default ) \family default \family typewriter norm( \emph on A \emph default , \emph on p \emph default ) * norm(inv( \emph on A \emph default ), \emph on p \emph default ) \family default . \end_layout \begin_layout Description \family typewriter cond( \series bold \emph on A \series default \emph default ) \family default \family typewriter cond( \emph on A \emph default ,2) \family default . \end_layout \begin_layout Description \family typewriter diag( \emph on A \emph default , \emph on k \emph default ) \family default Si \family typewriter \emph on A \family default \emph default es vector, devuelve una matriz diagonal con elementos del vector en la diagonal, y de lo contrario devuelve un vector con los elementos de la diagonal de \family typewriter \emph on A \family default \emph default . \end_layout \begin_layout Description \family typewriter dot( \emph on x \emph default , \emph on y \emph default ) \family default Producto escalar hermitiano \begin_inset Formula $\langle\text{\emph{\texttt{y}}},\text{\emph{\texttt{x}}}\rangle$ \end_inset . \end_layout \begin_layout Description \family typewriter [ \emph on V \emph default , \emph on lambda \emph default ]=eig( \emph on A \emph default ) \family default Devuelve una matriz diagonal \family typewriter \emph on lambda \family default \emph default en la que los elementos de la diagonal son los valores propios de \family typewriter \emph on A \family default \emph default y una matriz \family typewriter \emph on V \family default \emph default cuyas columnas son los vectores propios correspondientes. \end_layout \begin_layout Description \family typewriter eye( \emph on n \emph default ) \family default Matriz identidad de tamaño \family typewriter \emph on n \family default \emph default . \end_layout \begin_layout Description \family typewriter inv( \emph on A \emph default ) \family default Inversa de la matriz cuadrada no singular \family typewriter \emph on A \family default \emph default . \end_layout \begin_layout Description \family typewriter linspace( \emph on start \emph default , \emph on end \emph default , \emph on n \emph default ) \family default Vector de \family typewriter \emph on n \family default \emph default puntos equiespaciados de \family typewriter \emph on start \family default \emph default a \family typewriter \emph on end \family default \emph default . \end_layout \begin_layout Description \family typewriter norm( \emph on A \series medium \emph default , \emph on p \emph default ) \family default Norma \family typewriter \series default \emph on p \family default \emph default de \family typewriter \emph on A \family default \emph default , matricial o vectorial según corresponda, donde \family typewriter \emph on p \family default \emph default es un entero positivo o \family typewriter Inf \family default . \end_layout \begin_layout Description \family typewriter norm( \emph on A \emph default ) \family default \family typewriter norm( \emph on A \emph default ,2) \family default . \end_layout \begin_layout Description \family typewriter rand( \emph on m \emph default , \emph on n \emph default ) \family default Matriz de \family typewriter \emph on m \family default \emph default filas y \family typewriter \emph on n \family default \emph default columnas con elementos aleatorios entre 0 y 1. \end_layout \begin_layout Description \family typewriter [ \emph on U \emph default , \emph on S \emph default , \emph on V \emph default ]=svd( \emph on A \emph default ) \family default Devuelve dos matriz ortogonales \family typewriter \emph on U \family default \emph default y \family typewriter \emph on V \family default \emph default y una diagonal \family typewriter \emph on S \family default \emph default tales que \begin_inset Formula $\text{\emph{\texttt{A}}}=\text{\emph{\texttt{U}}}\text{\emph{\texttt{S}}}\text{\emph{\texttt{V}}}^{*}$ \end_inset . \end_layout \begin_layout Description \family typewriter trace( \emph on A \emph default ) \family default Traza de \family typewriter \emph on A \family default \emph default . \end_layout \begin_layout Description \family typewriter tril( \emph on A \emph default , \emph on k \emph default ) \family default Matriz como \family typewriter \emph on A \family default \emph default pero con los elementos \begin_inset Formula $(i,j)$ \end_inset con \begin_inset Formula $j-i>\text{\emph{\texttt{k}}}$ \end_inset a 0. \end_layout \begin_layout Description \family typewriter tril( \emph on A \emph default ) \family default \family typewriter tril( \emph on A \emph default ,0) \family default , matriz triangular inferior. \end_layout \begin_layout Description \family typewriter triu( \emph on A \emph default , \emph on k \emph default ) \family default Matriz como \family typewriter \emph on A \family default \emph default pero con los elementos \begin_inset Formula $(i,j)$ \end_inset con \begin_inset Formula $i-j>\text{\emph{\texttt{k}}}$ \end_inset a 0. \end_layout \begin_layout Description \family typewriter triu( \emph on A \series bold \emph default ) \family default \series default \family typewriter triu( \emph on A \emph default ,0) \family default , matriz triangular superior. \end_layout \begin_layout Description \family typewriter zeros( \emph on m \emph default , \emph on n \emph default ) \family default Matriz nula de \family typewriter \emph on m \family default \emph default filas y \family typewriter \emph on n \family default \emph default columnas. \end_layout \begin_layout Section Descomposiciones matriciales \end_layout \begin_layout Description \family typewriter [ \series bold \emph on L \emph default , \emph on U \emph default , \emph on P \emph default , \series default \emph on Q \series bold \emph default ]=lu( \series default \emph on A \emph default ) \family default Descomposición de Gauss de \family typewriter \series bold \emph on A \family default \series default \emph default con elección de pivote total. \end_layout \begin_layout Description \family typewriter [ \emph on L \emph default , \emph on U \emph default , \emph on P \emph default ]=lu( \emph on A \emph default ) \family default Descomposición de Gauss de \family typewriter \emph on A \family default \emph default con elección de pivote parcial. \end_layout \end_body \end_document