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pammacdotnet/endomorfismo + autovalores.html

Created January 15, 2023 15:58
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endomorfismo + autovalores.html
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN" "http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta name="GENERATOR" content="LyX 2.3.7" />
<meta http-equiv="Content-type" content="text/html;charset=UTF-8" />
<title>Autovalores desde endomorfismo</title>
<style type='text/css'>
/* Layout-provided Styles */
h1.title {
font-size: x-large;
margin-bottom: 1ex;
text-align: center;
}
div.standard {
margin-bottom: 2ex;
}
div.plain_layout {
text-align: left;
}
div.Shadowbox { border: solid gray medium; border-bottom: solid black 1ex; border-right: solid black 1ex; padding: 0.5ex; }
</style>
</head>
<body dir="auto">
<h1 class="title" id='magicparlabel-1'>Autovalores desde endomorfismo</h1>
<div class="standard" id='magicparlabel-2'>Sea el endomorfismo de <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mrow>
<mstyle mathvariant='double-struck'><mi>R</mi>
</mstyle>
</mrow>
<mrow><mn>3</mn>
</mrow>
</msup>
</mrow></math>en <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mrow>
<mstyle mathvariant='double-struck'><mi>R</mi>
</mstyle>
</mrow>
<mrow><mn>3</mn>
</mrow>
</msup>
</mrow></math> definido por: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mrow><mi>f</mi><mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</mo>
<mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi>
</mrow>
<mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</mo>
<mo>=</mo><mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</mo>
<mrow>
<mover>
<mrow>
<mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi>
</mrow>
</mrow><mo stretchy="true">&OverBrace;</mo>
</mover><mo>,</mo>
<munder>
<mrow>
<mrow><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi>
</mrow>
</mrow><mo stretchy="true">&UnderBrace;</mo>
</munder><mo>,</mo>
<mover>
<mrow>
<mrow><mn>3</mn><mi>z</mi>
</mrow>
</mrow><mo stretchy="true">&OverBrace;</mo>
</mover>
</mrow>
<mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</mo>
<mo>,</mo>
</mrow>
</mrow></math></div>
<div class="standard" id='magicparlabel-89'>calcule sus autovalores.</div>
<div class="standard" id='magicparlabel-97'><hr />
</div>
<div class="standard" id='magicparlabel-120'>Reexpresamos en endomorfismo con suma de vectores:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mrow><mi>f</mi><mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</mo>
<mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi>
</mrow>
<mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</mo>
<mo>=</mo><mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</mo>
<mrow><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn>
</mrow>
<mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</mo>
<mo>+</mo><mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</mo>
<mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>0</mn>
</mrow>
<mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</mo>
<mo>+</mo><mo form='prefix' fence='true' stretchy='true' symmetric='true'>(</mo>
<mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn>
</mrow>
<mo form='postfix' fence='true' stretchy='true' symmetric='true'>)</mo>
<mo>,</mo>
</mrow>
</mrow></math></div>
<div class="standard" id='magicparlabel-185'>de manera que podemos expresarlo matricialmente como <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>F</mi>
</mrow></math>:</div>
<div class="standard" id='magicparlabel-198'><math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mrow><mi>F</mi><mo>=</mo><mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>(</mo>
<mtable>
<mtr>
<mtd><mn>1</mn>
</mtd>
<mtd><mn>0</mn>
</mtd>
<mtd><mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd><mn>1</mn>
</mtd>
<mtd><mn>2</mn>
</mtd>
<mtd><mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd><mn>1</mn>
</mtd>
<mtd><mn>1</mn>
</mtd>
<mtd><mn>3</mn>
</mtd>
</mtr>
</mtable><mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>)</mo><mn>.</mn>
</mrow>
</mrow></math></div>
<div class="standard" id='magicparlabel-206'>Los autovalores de <math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>F</mi>
</mrow></math>:</div>
<div class="standard" id='magicparlabel-218'><math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>&ctdot;
</mi>
</mrow></math></div>
<div class="standard" id='magicparlabel-222'>De manera que el resultado es:</div>
<div class="standard" id='magicparlabel-225'><div class='Shadowbox'><div class="plain_layout" id='magicparlabel-260'><math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mrow>
<msub>
<mrow><mi> &lambda; </mi>
</mrow>
<mrow><mn>1</mn>
</mrow>
</msub><mo>=</mo><mn>3</mn><mo>,</mo><mspace width="40px" />
<msub>
<mrow><mi> &lambda; </mi>
</mrow>
<mrow><mn>2</mn>
</mrow>
</msub><mo>=</mo><mn>2</mn><mo>,</mo><mspace width="40px" />
<msub>
<mrow><mi> &lambda; </mi>
</mrow>
<mrow><mn>3</mn>
</mrow>
</msub><mo>=</mo><mn>1</mn>
</mrow>
</mrow></math></div>
</div></div>
</body>
</html>
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