How to determine eigenvalues and eignevectors of a matrix in Excel
We would like to compute the eigenvalues and eigenvectors of the following matrix in Excel. BTW Excel does not have any native function that can do that.
\$\begin{bmatrix} 2 & -1 & 0\\-1 & 2 & -1\\0 & -1 & 2\end{bmatrix}\$
The characteristic polynomial of this matrix is : \$det( A - \lambda I)\$ qui donne \$(2 - \lambda)(2 - \sqrt{2} - \lambda)(2 + \sqrt{2} - \lambda)\$ .
The analytical eigenvalues solutions are: \$\lambda_1=2 - \sqrt{2}=0.58578644\$, \$\lambda_2=2 \$ and \$\lambda_3=2 + \sqrt{2}=3.41421356\$
The eigenvectors are :
\$\begin{Bmatrix} 1 \\ \sqrt{2}\\1\end{Bmatrix}\$, \$\begin{Bmatrix} 1 \\0\\-1\end{Bmatrix}\$ and \$\begin{Bmatrix} 1 \\-\sqrt{2}\\1\end{Bmatrix}\$
Procedure
1st method
We start by creating two columns, one for l (from 0 to 4) and the other for \$ det( A - \lambda I)\$
Then we plot \$det( A - \lambda I) = f(\lambda)\$
We can clearly see the three zeros of the plotted function. Eventually we can decrease the step around each solu…
\$\begin{bmatrix} 2 & -1 & 0\\-1 & 2 & -1\\0 & -1 & 2\end{bmatrix}\$
The characteristic polynomial of this matrix is : \$det( A - \lambda I)\$ qui donne \$(2 - \lambda)(2 - \sqrt{2} - \lambda)(2 + \sqrt{2} - \lambda)\$ .
The analytical eigenvalues solutions are: \$\lambda_1=2 - \sqrt{2}=0.58578644\$, \$\lambda_2=2 \$ and \$\lambda_3=2 + \sqrt{2}=3.41421356\$
The eigenvectors are :
\$\begin{Bmatrix} 1 \\ \sqrt{2}\\1\end{Bmatrix}\$, \$\begin{Bmatrix} 1 \\0\\-1\end{Bmatrix}\$ and \$\begin{Bmatrix} 1 \\-\sqrt{2}\\1\end{Bmatrix}\$
Procedure
1st method
We start by creating two columns, one for l (from 0 to 4) and the other for \$ det( A - \lambda I)\$
Then we plot \$det( A - \lambda I) = f(\lambda)\$
We can clearly see the three zeros of the plotted function. Eventually we can decrease the step around each solu…
