Find the spectral decomposition of the \mathbf{S}. That is, find the eigenvalues and their corresponding eigenvectors.
Enter the eigenvalues in decreasing order (so \lambda _1>\lambda _2.)
\lambda ^{(1)}=\quad
unanswered
\lambda ^{(2)}=\quad
unanswered
Find the eigenvectors \mathbf v_{\lambda _1} and \mathbf v_{\lambda _2}. (All scalar multiples will be accepted)
\mathbf v_{\lambda _1}=\quad
unanswered
{::}
\mathbf v_{\lambda _2}=\quad
unanswered
{::}
Without having the matrix \mathbf{S}, it is not possible to determine its eigenvalues and eigenvectors. Please provide the matrix \mathbf{S} in order to solve the problem.
To find the spectral decomposition of matrix \mathbf{S}, we need to find its eigenvalues and their corresponding eigenvectors.
First, let's find the eigenvalues by solving the characteristic equation:
det(\mathbf{S} - \lambda \mathbf{I}) = 0
where \mathbf{I} is the identity matrix.
Once we find the eigenvalues, we can find the eigenvectors by solving the following equation for each eigenvalue:
(\mathbf{S} - \lambda \mathbf{I}) \mathbf{v} = \mathbf{0}
Let's go step by step:
Step 1: Find the eigenvalues
Start by subtracting \lambda from the diagonal of \mathbf{S}:
\mathbf{S} - \lambda \mathbf{I} =
\begin{bmatrix}
s_{11}-\lambda & s_{12} & s_{13} & \ldots \\
s_{21} & s_{22}-\lambda & s_{23} & \ldots \\
s_{31} & s_{32} & s_{33}-\lambda & \ldots \\
\vdots & \vdots & \vdots & \ddots \\
\end{bmatrix}
Take the determinant of the resulting matrix and set it equal to zero:
det(\mathbf{S} - \lambda \mathbf{I}) = 0
Solve this equation to find the eigenvalues. Enter the eigenvalues you found in decreasing order as follows:
\lambda ^{(1)} = unanswered
\lambda ^{(2)} = unanswered
Step 2: Find the eigenvectors
For each eigenvalue, substitute it back into the equation (\mathbf{S} - \lambda \mathbf{I}) \mathbf{v} = \mathbf{0} and solve for the corresponding eigenvectors.
For \lambda ^{(1)}, substitute it into the equation:
(\mathbf{S} - \lambda ^{(1)} \mathbf{I}) \mathbf{v} = \mathbf{0}
Solve this equation to find the eigenvector \mathbf{v}_{\lambda ^{(1)}}. Enter the eigenvector here:
\mathbf{v}_{\lambda ^{(1)}} = unanswered
For \lambda ^{(2)}, substitute it into the equation:
(\mathbf{S} - \lambda ^{(2)} \mathbf{I}) \mathbf{v} = \mathbf{0}
Solve this equation to find the eigenvector \mathbf{v}_{\lambda ^{(2)}}. Enter the eigenvector here:
\mathbf{v}_{\lambda ^{(2)}} = unanswered
To find the spectral decomposition of a matrix, we need to find its eigenvalues and their corresponding eigenvectors. Here's how you can find the eigenvalues and eigenvectors:
1. Start with the given matrix \mathbf{S}.
2. Find the characteristic polynomial by subtracting the identity matrix multiplied by a scalar λ from \mathbf{S}. The characteristic polynomial is given by det(\mathbf{S} - \lambda \mathbf{I}), where det() denotes the determinant, \mathbf{I} is the identity matrix, and λ is the eigenvalue we are solving for.
3. Set the characteristic polynomial equal to zero and solve for λ. The solutions to this equation are the eigenvalues of \mathbf{S}.
4. Once you have the eigenvalues, substitute each eigenvalue into the equation (\mathbf{S} - \lambda \mathbf{I})\mathbf{v} = \mathbf{0} to find the corresponding eigenvectors \mathbf{v}. In this equation, \mathbf{v} is the eigenvector corresponding to the eigenvalue λ.
5. Normalize the eigenvectors if needed, i.e., divide each eigenvector by its magnitude to get a unit vector.
6. Repeat steps 4 and 5 for each eigenvalue.
7. Sort the eigenvalues in decreasing order and label them as \lambda ^{(1)}, \lambda ^{(2)}, etc.
8. Finally, enter the eigenvalues in decreasing order and the corresponding eigenvectors as requested.
Note: The eigenvectors are not unique, and any scalar multiple of an eigenvector is also an eigenvector corresponding to the same eigenvalue.
Follow these steps to calculate the spectral decomposition of the matrix \mathbf{S}.