document.write( "Question 1189592: A transformation matrix is given by M= [4 -1]
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Algebra.Com's Answer #821025 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "A transformation matrix is given by M= [4 -1]
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\n" ); document.write( "Show that the transformation matrix has no invariant line using the equation of line y=mx
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document.write( "Had the invariant line exist, the given matrix would have real eigenvalue.\r\n" );
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document.write( "But it is easy to show that all its eigenvalues are complex numbers, not real.\r\n" );
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document.write( "Indeed, the characteristic matrix is  .\r\n" );
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document.write( "Its characteristic equation is  \r\n" );
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document.write( "    (4-x)*(3-x) + 2 = 0,\r\n" );
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document.write( "or\r\n" );
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document.write( "    12 - 3x - 4x + x^2 + 2 = 0\r\n" );
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document.write( "    x^2 - 7x + 14 = 0.    (*)\r\n" );
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document.write( "The discriminant of this quadratic equation is  d = (-7)^2 - 4*1*14 = 49 - 56 = -7  less than zero.\r\n" );
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document.write( "It means that the characteristic equation has no real solution/solutions;\r\n" );
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document.write( "so, the original matrix has no real eigenvalues.\r\n" );
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document.write( "Thus the statement is proved.\r\n" );
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For your better understanding

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\n" ); document.write( "\n" ); document.write( "        Since this matrix has complex value eigenvalues, the associated transformation of a plane \r
\n" ); document.write( "\n" ); document.write( "        is stretching-compression combined with rotation.\r
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\n" ); document.write( "\n" ); document.write( "        Therefore, there is no invariant lines: rotation excludes such lines.\r
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