- #1

Mindscrape

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The problem is to solve the differential equation where

[tex]\mathbf{x'} = \left(

\begin{array}{cc}

1 & -5\\

1 & -3

\end{array}

\right)

\mathbf{x}

[/tex]

given that

[tex] \mathbf{x(0)} = \left( \begin{array}{cc} 5 \\ 4 \end{array} \right) [/tex]

The eigenvalues are easy to find, and they are:

[tex] \lambda = 1 + i[/tex]

and

[tex] \lambda = 1 - i[/tex]

I am having trouble reducing the complex matrix with Gauss-Jordan elimination, and was wondering if I could get some pointers. Here is what I have done:

By pluging in the first eigenvalue the matrix to reduce is

[tex] \left( \begin{array}{cc}

-i & -5\\

1 & -4 - i

\end{array}

\right)[/tex]

then I figure that I should multiply by i on the top and divide by -4-i on the bottom so that the pivots are 1 and real:

This makes

[tex] \left( \begin{array}{cc}

1 & -5i\\

\frac{1}{-4-i} & 1

\end{array}

\right)[/tex]

I tried to take it from here, but I wasn't able to get anywhere. I tried complex conjugates in various places, and multiplying i here and there, but no luck. Most of the difficulty I am having is just reducing the second row to the zero row. I can get a zero at (2,1), but getting (2,2) to zero is proving difficult.

[tex]\mathbf{x'} = \left(

\begin{array}{cc}

1 & -5\\

1 & -3

\end{array}

\right)

\mathbf{x}

[/tex]

given that

[tex] \mathbf{x(0)} = \left( \begin{array}{cc} 5 \\ 4 \end{array} \right) [/tex]

The eigenvalues are easy to find, and they are:

[tex] \lambda = 1 + i[/tex]

and

[tex] \lambda = 1 - i[/tex]

I am having trouble reducing the complex matrix with Gauss-Jordan elimination, and was wondering if I could get some pointers. Here is what I have done:

By pluging in the first eigenvalue the matrix to reduce is

[tex] \left( \begin{array}{cc}

-i & -5\\

1 & -4 - i

\end{array}

\right)[/tex]

then I figure that I should multiply by i on the top and divide by -4-i on the bottom so that the pivots are 1 and real:

This makes

[tex] \left( \begin{array}{cc}

1 & -5i\\

\frac{1}{-4-i} & 1

\end{array}

\right)[/tex]

I tried to take it from here, but I wasn't able to get anywhere. I tried complex conjugates in various places, and multiplying i here and there, but no luck. Most of the difficulty I am having is just reducing the second row to the zero row. I can get a zero at (2,1), but getting (2,2) to zero is proving difficult.

Last edited: