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908 lines
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Eigenvector and Eigenvalue</h1>
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<p>They have many uses!</p>
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<p>A simple example is that an eigenvector <b>does not change direction</b> in a transformation:</p>
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<p class="center"><img src="images/eigen-transform.svg" alt="Eigenvector in transformation" height="244" width="548"></p>
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<p>How do we find that vector?</p>
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<h2>The Mathematics Of It</h2>
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<p>For a square matrix <b>A</b>, an Eigenvector and Eigenvalue make this equation true:</p>
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<p class="center"><img src="images/eigenvalue.svg" alt="A times x = lambda times x" height="109" width="275"></p>
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<p>Let us see it in action:</p>
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<div class="example">
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<h3>Example: For this matrix<br>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>−6</div>
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<div>3</div>
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<div>4</div>
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<div>5</div>
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</div>
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</div>
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</div>
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<!-- [-6,3~4,5] --><br>
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an eigenvector is<br>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols1">
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<div>1</div>
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<div>4</div>
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</div>
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</div>
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</div>
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<!-- [1~4] --><br>
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with a matching eigenvalue of <span class="larger">6</span></h3>
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<p>Let's do some <a href="matrix-multiplying.html">matrix multiplies</a> to see if that is true.</p>
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<p><span class="center large">Av</span> gives us:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2" style="color:#850;">
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<div>−6</div>
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<div>3</div>
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<div>4</div>
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<div>5</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1" style="color:#f80;">
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<div>1</div>
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<div>4</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>−6×1+3×4</div>
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<div>4×1+5×4</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>6</div>
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<div>24</div>
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</div>
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</div>
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</div>
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<!-- [-6,3~4,5][1~4] = [-6*1+3*4~4*1+5*4] = [6~24] -->
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<p><span class="center large">λv</span> gives us :</p>
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<div style="text-align: center;">
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<div class="txt" style="width:35px"> </div>
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<div class="txt" style="color:#05c;">6</div>
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<div class="mat">
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<div class="cols1" style="color:#f80;">
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<div>1</div>
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<div>4</div>
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</div>
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</div>
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<div class="txt" style="width:205px">=</div>
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<div class="mat">
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<div class="cols1">
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<div>6</div>
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<div>24</div>
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</div>
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</div>
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</div>
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<!-- 6[1~4] = [6~24] -->
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<p>Yes they are equal! </p>
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<p>So we get <span class="large">Av = λv</span> as promised.</p>
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</div>
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<p>Notice how we multiply a <b>matrix</b> by a <b>vector</b> and get the same result as when we multiply a <b>scalar</b> (just a number) by that <b>vector</b>.</p>
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<h2>How do we find these <span class="center">eigen things?</span></h2>
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<p>We start by finding the <b>eigenvalue</b>. We know this equation must be true:</p>
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<p class="center large">Av = λv</p>
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<p>Next we put in an <a href="matrix-types.html">identity matrix</a> so we are dealing with matrix-vs-matrix:</p>
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<p class="center large">Av = λIv</p>
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<p>Bring all to left hand side:</p>
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<p class="center large">Av − λIv = 0</p>
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<p>If <b>v</b> is non-zero then we can (hopefully) solve for <span class="center large">λ</span> using just the <a href="matrix-determinant.html">determinant</a>:</p>
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<p class="center large">| A − λI | = 0</p>
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<p>Let's try that equation on our previous example:</p>
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<div class="example">
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<h3>Example: Solve for <span class="center large">λ</span></h3>
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<p>Start with <span class="larger">| A − λI | = 0</span></p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="width:10px;">
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<div class="larger" style="transform: scaleY(3) translateY(-1px); vertical-align:top;"><b>|</b></div></td>
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<td>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>−6</div>
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<div>3</div>
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<div>4</div>
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<div>5</div>
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</div>
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</div>
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<div class="txt">− λ</div>
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<div class="mat">
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<div class="cols2">
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<div>1</div>
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<div>0</div>
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<div>0</div>
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<div>1</div>
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</div>
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</div>
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</div>
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<!-- [-6,3~4,5] - λ[1,0~0,1] --></td>
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<td style="width:10px;">
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<div class="larger" style="transform: scaleY(3) translateY(-1px); vertical-align:top;"><b>|</b></div></td>
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<td><span class="larger"> = 0</span></td>
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</tr>
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</tbody></table>
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<p>Which is:</p>
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<div style="text-align: center;">
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<div class="det">
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<div class="cols2">
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<div>−6−λ</div>
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<div>3</div>
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<div>4</div>
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<div>5−λ</div>
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</div>
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</div>
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<span class="larger">= 0</span>
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</div>
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<p><!-- [-6-λ,3~4,5-λ] = 0 -->
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Calculating that determinant gets:</p>
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<p class="center larger">(−6−λ)(5−λ) − 3×4 = 0</p>
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<p>Which simplifies to this <a href="quadratic-equation.html">Quadratic Equation</a>:</p>
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<p class="center larger">λ<sup>2</sup> + λ − 42 = 0</p>
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<p>And <a href="../quadratic-equation-solver.html">solving it</a> gets:</p>
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<p class="center larger">λ = −7 or 6</p>
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<p>And yes, there are <b>two</b> possible eigenvalues.</p>
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</div>
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<p>Now we know <b>eigenvalues</b>, let us find their matching <b>eigenvectors</b>.</p>
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<div class="example">
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<h3>Example (continued): Find the Eigenvector for the Eigenvalue <b>λ = 6</b>:</h3>
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<p>Start with:</p>
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<p class="center large">Av = λv</p>
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Put in the values we know:<br>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>−6</div>
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<div>3</div>
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<div>4</div>
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<div>5</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt">= 6</div>
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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</div>
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<!-- [-6,3~4,5][x~y] = 6[x~y] -->
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<p>After multiplying we get these two equations:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">−6x + 3y </td>
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<td style="text-align:center; width:25px;">=</td>
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<td>6x</td>
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</tr>
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<tr>
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<td style="text-align:right;">4x + 5y </td>
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<td style="text-align:center; width:25px;">=</td>
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<td>6y</td>
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</tr>
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</tbody></table>
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<p>Bringing all to left hand side:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">−12x + 3y </td>
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<td style="text-align:center; width:25px;">=</td>
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<td>0</td>
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</tr>
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<tr>
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<td style="text-align:right;">4x − 1y </td>
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<td style="text-align:center; width:25px;">=</td>
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<td>0</td>
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</tr>
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</tbody></table>
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<p><i>Either</i> equation reveals that <b>y = 4x</b>, so the <b>eigenvector</b> is any non-zero multiple of this:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols1">
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<div>1</div>
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<div>4</div>
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</div>
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</div>
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</div>
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<!-- [1~4] -->
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<p>And we get the solution shown at the top of the page:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2" style="color:#850;">
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<div>−6</div>
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<div>3</div>
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<div>4</div>
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<div>5</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1" style="color:#f80;">
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<div>1</div>
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<div>4</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>−6×1+3×4</div>
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<div>4×1+5×4</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>6</div>
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<div>24</div>
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</div>
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</div>
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</div>
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<!-- [-6,3~4,5][1~4] = [-6*1+3*4~4*1+5*4] = [6~24] -->
|
||
<p class="center">... and also ...</p>
|
||
<div style="text-align: center;">
|
||
<div class="txt" style="width:35px"> </div>
|
||
<div class="txt" style="color:#05c;">6</div>
|
||
<div class="mat">
|
||
<div class="cols1" style="color:#f80;">
|
||
<div>1</div>
|
||
<div>4</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt" style="width:205px">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>6</div>
|
||
<div>24</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- 6[1~4] = [6~24] -->
|
||
<p>So <span class="large">Av = λv</span>, and we have success!<span class="large"> <br></span></p>
|
||
</div>
|
||
<p>Now it is <b>your turn</b> to find the eigenvector for the other eigenvalue of <span class="larger">−7</span></p>
|
||
|
||
|
||
<h2>Why?</h2>
|
||
|
||
<p>What is the purpose of these?</p>
|
||
<p>One of the cool things is we can use <a href="matrix-introduction.html">matrices</a> to do <a href="matrix-transform.html">transformations</a> in space, which is used a lot in computer graphics.</p>
|
||
<p>In that case the eigenvector is "the direction that doesn't change direction" !</p>
|
||
<p>And the eigenvalue is the scale of the stretch:</p>
|
||
<ul>
|
||
<li><b>1</b> means no change,</li>
|
||
<li><b>2</b> means doubling in length,</li>
|
||
<li><b>−1</b> means pointing backwards along the eigenvalue's direction</li>
|
||
<li>etc</li>
|
||
</ul>
|
||
<p>There are also many applications in physics, etc.</p>
|
||
|
||
|
||
<h2>Why "Eigen"</h2>
|
||
|
||
<div class="words">
|
||
<p>Eigen is a German word meaning "own" or "typical"</p>
|
||
<p class="center"><i>"das ist ihnen <b>eigen</b>"</i> is German for <i>"that is <b>typical</b> of them"</i></p>
|
||
</div>
|
||
<p>Sometimes in English we use the word "characteristic", so an eigenvector can be called a "characteristic vector".</p>
|
||
|
||
|
||
<h2>Not Just Two Dimensions</h2>
|
||
|
||
<p>Eigenvectors work perfectly well in 3 and higher dimensions.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: find the eigenvalues for this 3x3 matrix:
|
||
|
||
|
||
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols3">
|
||
<div>2</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>5</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>3</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [2,0,0~0,4,5~0,4,3] --></h3>
|
||
<p>First calculate <span class="center large">A − λI</span>:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols3">
|
||
<div>2</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>5</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>3</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">− λ</div>
|
||
<div class="mat">
|
||
<div class="cols3">
|
||
<div>1</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>1</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols3">
|
||
<div>2−λ</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>4−λ</div>
|
||
<div>5</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>3−λ</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [2,0,0~0,4,5~0,4,3] - λ[1,0,0~0,1,0~0,0,1] = [2-λ,0,0~0,4-λ,5~0,4,3-λ] -->
|
||
<p>Now the determinant should equal zero:</p>
|
||
<div style="text-align: center;">
|
||
<div class="det">
|
||
<div class="cols3">
|
||
<div>2−λ</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>4−λ</div>
|
||
<div>5</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>3−λ</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">= 0</div>
|
||
</div>
|
||
<!-- [2-λ,0,0~0,4-λ,5~0,4,3-λ] = 0 -->
|
||
<p>Which is:</p>
|
||
<p class="center large">(2−λ) [ (4−λ)(3−λ) − 5×4 ] = 0</p>
|
||
<p>This ends up being a cubic equation, but just looking at it here we see one of the roots is <b>2</b> (because of 2−λ), and the part inside the square brackets is Quadratic, with <a href="../quadratic-equation-solver.html">roots</a> of <b>−1</b> and <b>8</b>.</p>
|
||
<p>So the Eigenvalues are <b>−1</b>, <b>2</b> and <b>8</b></p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example (continued): find the Eigenvector that matches the Eigenvalue <b>−1</b></h3>
|
||
<p>Put in the values we know:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols3">
|
||
<div>2</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>5</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>3</div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>x</div>
|
||
<div>y</div>
|
||
<div>z</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">= −1</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>x</div>
|
||
<div>y</div>
|
||
<div>z</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [2,0,0~0,4,5~0,4,3][x~y~z] = 8[x~y~z] -->
|
||
<p>After multiplying we get these equations:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;">2x </td>
|
||
<td style="text-align:center; width:25px;">=</td>
|
||
<td>−x</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">4y + 5z </td>
|
||
<td style="text-align:center; width:25px;">=</td>
|
||
<td>−y</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">4y + 3z </td>
|
||
<td style="text-align:center;">=</td>
|
||
<td>−z</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>Bringing all to left hand side:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;">3x</td>
|
||
<td style="text-align:center; width:25px;">=</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">5y + 5z </td>
|
||
<td style="text-align:center; width:25px;">=</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">4y + 4z </td>
|
||
<td style="text-align:center;">=</td>
|
||
<td>0</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>So <b>x = 0</b>, and <b>y = −z</b> and so the <b>eigenvector</b> is any non-zero multiple of this:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>0</div>
|
||
<div>1</div>
|
||
<div>−1</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [0~1~-1] -->
|
||
<p>TEST <span class="large">Av</span>:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols3">
|
||
<div>2</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>5</div>
|
||
<div>0</div>
|
||
<div>4</div>
|
||
<div>3</div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>0</div>
|
||
<div>1</div>
|
||
<div>−1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>0</div>
|
||
<div>4−5</div>
|
||
<div>4−3</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>0</div>
|
||
<div>−1</div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [2,0,0~0,4,5~0,4,3][0~1~-1] = [0~4-5~4-3] = [0~-1~1] -->
|
||
<p>And <span class="large">λv</span>:</p>
|
||
<div style="text-align: center;">
|
||
<div class="txt">−1</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>0</div>
|
||
<div>1</div>
|
||
<div>−1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>0</div>
|
||
<div>−1</div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- -1[0~1~-1] = [0~-1~1] -->
|
||
<p>So <span class="large">Av = λv</span>, yay!</p>
|
||
<p>(You can try your hand at the eigenvalues of <b>2</b> and <b>8</b>)</p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Rotation</h2>
|
||
|
||
<p>Back in the 2D world again, this matrix will do a rotation by θ:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>cos(θ)</div>
|
||
<div>−sin(θ)</div>
|
||
<div>sin(θ)</div>
|
||
<div>cos(θ)</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [cos(θ),-sin(θ)~sin(θ),cos(θ)] = 0 -->
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Rotate by 30°</h3>
|
||
<p>cos(30°) = <span class="intbl"><em>√3</em><strong>2</strong></span> and sin(30°) = <span class="intbl"><em>1</em><strong>2</strong></span>, so:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>cos(30°)</div>
|
||
<div>−sin(30°)</div>
|
||
<div>sin(30°)</div>
|
||
<div>cos(30°)</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>−1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [cos(30°),-sin(30°)~sin(30°),cos(30°)] = [sqr3/2,-1/2~1/2,sqr3/2] -->
|
||
<p>But if we <b>rotate all points</b>, what is the "direction that doesn't change direction"?</p>
|
||
<p class="center"><img src="images/transform-rotate.svg" alt="A Rotation Transformation" height="241" width="212"></p>
|
||
<p> </p>
|
||
<p>Let us work through the mathematics to find out:</p>
|
||
<p>First calculate <span class="center large">A − λI</span>:</p>
|
||
<p><br></p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>−1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">− λ</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>1</div>
|
||
<div>0</div>
|
||
<div>0</div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span>−λ</div>
|
||
<div><span class="intbl"><em>−1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span>−λ</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt"></div>
|
||
</div>
|
||
<!-- [sqr3/2,-1/2~1/2,sqr3/2] - λ[1,0~0,1] = [sqr3/2 - λ,-1/2~1/2,sqr3/2 - λ] -->
|
||
<p>Now the determinant should equal zero:</p>
|
||
<div style="text-align: center;">
|
||
<div class="det">
|
||
<div class="cols2">
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span>−λ</div>
|
||
<div><span class="intbl"><em>−1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span>−λ</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">= 0</div>
|
||
</div>
|
||
<!-- [sqr3/2 - λ,-1/2~1/2,sqr3/2 - λ] = 0 -->
|
||
<p>Which is:</p>
|
||
<p class="center large">(<span class="intbl"><em>√3</em><strong>2</strong></span>−λ)(<span class="intbl"><em>√3</em><strong>2</strong></span>−λ) − (<span class="intbl"><em>−1</em><strong>2</strong></span>)(<span class="intbl"><em>1</em><strong>2</strong></span>) = 0</p>
|
||
<!-- ( sqr3/2 - λ )( sqr3/2 - λ ) - ( -1/2 )( 1/2 ) = 0 -->
|
||
<p>Which becomes this Quadratic Equation:</p>
|
||
<p class="center large">λ<sup>2</sup> − (√3)λ + 1 = 0</p>
|
||
<!-- λ^2 - (sqr3)λ + 1 = 0 -->
|
||
<p>Whose roots are:</p>
|
||
<p class="center large">λ = <span class="intbl"><em>√3</em><strong>2</strong></span> ± <span class="intbl"><em><i><b>i</b></i></em><strong>2</strong></span></p>
|
||
<!-- λ = sqr3/2 +- i/2 -->
|
||
<p>The eigenvalues are complex!</p>
|
||
<p>I don't know how to show you that on a graph, but we still get a solution.</p>
|
||
|
||
<h3>Eigenvector</h3>
|
||
<p>So, what is an eigenvector that matches, say, the <span class="intbl"><em>√3</em><strong>2</strong></span> + <span class="intbl"><em><i><b>i</b></i></em><strong>2</strong></span> root?</p>
|
||
<p>Start with:</p>
|
||
<p class="center large">Av = λv</p>
|
||
Put in the values we know:<br>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>−1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>x</div>
|
||
<div>y</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">= (<span class="intbl"><em>√3</em><strong>2</strong></span> + <span class="intbl"><em><i><b>i</b></i></em><strong>2</strong></span>)</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>x</div>
|
||
<div>y</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [sqr3/2,-1/2~1/2,sqr3/2][x~y] = ( sqr3/2 + i/2 ) [x~y] -->
|
||
<p>After multiplying we get these two equations:</p>
|
||
<p class="center large"><span class="intbl"><em>√3</em><strong>2</strong></span>x − <span class="intbl"><em>1</em><strong>2</strong></span>y = <span class="intbl"><em>√3</em><strong>2</strong></span>x + <span class="intbl"><em><b><i>i</i></b></em><strong>2</strong></span>x</p>
|
||
<!-- sqr3/2 x - 1/2 y = sqr3/2 x + i/2 x -->
|
||
<p class="center large"><span class="intbl"><em>1</em><strong>2</strong></span>x + <span class="intbl"><em>√3</em><strong>2</strong></span>y = <span class="intbl"><em>√3</em><strong>2</strong></span>y + <span class="intbl"><em><b><i>i</i></b></em><strong>2</strong></span>y</p>
|
||
<!-- 1/2 x + sqr3/2 y = sqr3/2 y + i/2 y -->
|
||
<p>Which simplify to:</p>
|
||
<p class="center large">−y = <b><i>i</i></b>x</p>
|
||
<p class="center large">x = <b><i>i</i></b>y</p>
|
||
<p>And the solution is any non-zero multiple of:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div><b><i>i</i></b></div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [i~1] -->
|
||
<p>or</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>−<b><i>i</i></b></div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [-i~1] -->
|
||
<p><b>Wow, such a simple answer!</b></p>
|
||
<p><i>Is this just because we chose 30°? Or does it work for any rotation matrix? I will let you work that out! Try another angle, or better still use "cos(θ)" and "sin(θ)".</i></p>
|
||
<p> </p>
|
||
<p>Oh, and let us <b>check</b> at least one of those solutions:</p>
|
||
<div style="text-align: center;">
|
||
<div class="txt"></div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>−1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div><b><i>i</i></b></div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div><b><i>i</i></b><span class="intbl"><em>√3</em><strong>2</strong></span> − <span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em><b><i>i</i></b></em><strong>2</strong></span> + <span class="intbl"><em>√3</em><strong>2</strong></span></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [sqr3/2,-1/2~1/2,sqr3/2][i~1] = [i sqr3/2 - 1/2~i/2 + sqr3/2] -->
|
||
<p>Does it match this?</p>
|
||
<div style="text-align: center;">
|
||
<div class="txt">(<span class="intbl"><em>√3</em><strong>2</strong></span> + <span class="intbl"><em><b><i>i</i></b></em><strong>2</strong></span>)</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div><b><i>i</i></b></div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div><b><i>i</i></b><span class="intbl"><em>√3</em><strong>2</strong></span> − <span class="intbl"><em>1</em><strong>2</strong></span></div>
|
||
<div><span class="intbl"><em>√3</em><strong>2</strong></span> + <span class="intbl"><em><b><i>i</i></b></em><strong>2</strong></span></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- ( sqr3/2 + i/2 ) [i~1] = [i sqr3/2 - 1/2~sqr3/2 + i/2] -->
|
||
<p>Oh yes it does!</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">17820, 17821, 17804, 17805, 17806, 17807, 17814, 17818, 17819, 17808, 17809, 17810, 17811, 17812, 17813, 17815, 17816, 17817, 17822, 17823</div>
|
||
|
||
<div class="related">
|
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<a href="matrix-introduction.html">Introduction to Matrices</a>
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<a href="matrix-multiplying.html">Matrix Multiplication</a>
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<a href="matrix-index.html">Matrix Index</a>
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<a href="matrix-calculator.html">Matrix Calculator</a>
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<a href="index-2.html">Algebra 2 Index</a>
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