- #1
Benny
- 577
- 0
I'm experiencing difficulties trying to find the eigenvalues of the follow matrix. The hint is to use an elementary row operation to simplify C - \lambda I but I can't think of a suitable one to use or figure out whether a single row operation will actually make the calculations simpler.
<br /> C = \left[ {\begin{array}{*{20}c}<br /> {0.98} & {0.01} \\<br /> {0.02} & {0.99} \\<br /> \end{array}} \right]<br />
<br /> \det \left( {C - \lambda I} \right) = 0 \Rightarrow \left| {\begin{array}{*{20}c}<br /> {0.98 - \lambda } & {0.01} \\<br /> {0.02} & {0.99 - \lambda } \\<br /> \end{array}} \right| = 0<br />
Out of desperation, and having seen it being done once(not sure if it is correct) I decided to then subtract the first column from the second column.
<br /> \left| {\begin{array}{*{20}c}<br /> {0.98 - \lambda } & { - 0.97 + \lambda } \\<br /> {0.02} & {0.97 - \lambda } \\<br /> \end{array}} \right| = 0<br />
<br /> \left| {\begin{array}{*{20}c}<br /> {1 - \lambda } & 0 \\<br /> {0.02} & {0.97 - \lambda } \\<br /> \end{array}} \right| = 0<br />
I'm not even sure if subtracting columns from each other was a valid step. I know that subtracting rows is but I'm not sure about columns. I'm just wondering if that step was correct because if it is then I can get the eigenvalues from it fairly easily.
<br /> C = \left[ {\begin{array}{*{20}c}<br /> {0.98} & {0.01} \\<br /> {0.02} & {0.99} \\<br /> \end{array}} \right]<br />
<br /> \det \left( {C - \lambda I} \right) = 0 \Rightarrow \left| {\begin{array}{*{20}c}<br /> {0.98 - \lambda } & {0.01} \\<br /> {0.02} & {0.99 - \lambda } \\<br /> \end{array}} \right| = 0<br />
Out of desperation, and having seen it being done once(not sure if it is correct) I decided to then subtract the first column from the second column.
<br /> \left| {\begin{array}{*{20}c}<br /> {0.98 - \lambda } & { - 0.97 + \lambda } \\<br /> {0.02} & {0.97 - \lambda } \\<br /> \end{array}} \right| = 0<br />
<br /> \left| {\begin{array}{*{20}c}<br /> {1 - \lambda } & 0 \\<br /> {0.02} & {0.97 - \lambda } \\<br /> \end{array}} \right| = 0<br />
I'm not even sure if subtracting columns from each other was a valid step. I know that subtracting rows is but I'm not sure about columns. I'm just wondering if that step was correct because if it is then I can get the eigenvalues from it fairly easily.
