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
\mathbf v_{\lambda _2}=\quad
I'm sorry, but you haven't provided a matrix \mathbf{S}. Without the matrix, I cannot calculate the eigenvalues and eigenvectors. Please provide the matrix \mathbf{S}, and I will be happy to help you with the spectral decomposition.
To find the spectral decomposition of a matrix \mathbf{S}, we need to find the eigenvalues and their corresponding eigenvectors.
Let's denote the eigenvalues as \lambda ^{(1)} and \lambda ^{(2)}.
To find the eigenvalues, we solve the characteristic equation:
\det(\mathbf{S} - \lambda \mathbf{I}) = 0,
where \mathbf{I} is the identity matrix.
Once we find the eigenvalues \lambda ^{(1)} and \lambda ^{(2)}, we can find their corresponding eigenvectors by solving the equation:
(\mathbf{S} - \lambda \mathbf{I}) \mathbf{v} = 0,
where \mathbf{v} is the eigenvector.
Now, let's solve for the eigenvalues and eigenvectors:
1. Start with the characteristic equation:
\det(\mathbf{S} - \lambda \mathbf{I}) = 0.
2. Solve for \lambda by setting the determinant to zero. This will give us the eigenvalues \lambda ^{(1)} and \lambda ^{(2)}.
3. Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation:
(\mathbf{S} - \lambda \mathbf{I}) \mathbf{v} = 0.
4. Solve for \mathbf{v} to find the eigenvectors \mathbf v_{\lambda _1} and \mathbf v_{\lambda _2}. Note that all scalar multiples of the eigenvectors are also valid eigenvectors.
Please provide the matrix \mathbf{S} so that we can proceed with the calculations.
To find the spectral decomposition of matrix S, we need to find its eigenvalues and their corresponding eigenvectors. Here are the steps to do so:
1. Begin by solving the characteristic equation det(S - λI) = 0, where det denotes the determinant, S is the matrix in question, λ is the eigenvalue, and I is the identity matrix of the same size as S.
2. Solve the equation to find the eigenvalues λ1, λ2, λ3, ..., in decreasing order.
3. For each eigenvalue λ, solve the homogeneous equation (S - λI)v = 0 to find the corresponding eigenvectors v. Every nonzero solution to this equation is a valid eigenvector.
Now, let's solve the problem using the given matrix S:
1. Start by setting up the characteristic equation:
det(S - λI) = 0
2. Calculate the determinant of the matrix (S - λI) and set it equal to zero. This will give us the characteristic polynomial, which we can solve to find the eigenvalues. The eigenvalues will be the values of λ that satisfy this equation.
3. Once we have found the eigenvalues, we can find the corresponding eigenvectors by plugging each eigenvalue back into the equation (S - λI)v = 0 and solving it for v.
Now, please provide the matrix S so that we can proceed with finding its spectral decomposition.