\[ \begin{bmatrix} 4 & -5 & 7 & 0 \\ 0 & 3 & 1 & -5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]
This question aims to find the eigenvalues of an upper triangular matrix which are repeated according to their multiplicities.
The concept needed for this question includes eigenvalues and matrices. Eigenvalues are a set of scalar values that gives the importance or magnitude of the respective column of the matrix.
Expert Answer
The given matrix is an upper triangular matrix, which means that all the values below the main diagonal are zeros. The values above the main diagonal can be zero, but if all the values above and below the main diagonal are zero, then the matrix is called the diagonal matrix.
We know that the values at the main diagonal are all eigenvalues of the given matrix. The eigenvalues of the given matrix are:
\[ Eigenvalues\ =\ 4, 3, 1, 1 \]
We need to list these eigenvalues according to their multiplicities. The multiplicities of the eigenvalues are given as:
The eigenvector of $\lambda = 4$ is given as:
\[ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \]
\[ \lambda = 4 \longrightarrow multiplicity = 1 \]
The eigenvector of $\lambda = 3$ is given as:
\[ \begin{bmatrix} 5 \\ 1 \\ 0 \\ 0 \end{bmatrix} \]
\[ \lambda = 3 \longrightarrow multiplicity = 1 \]
The eigenvector of $\lambda = 1$ is given as:
\[ \begin{bmatrix} -\frac{19} {6} \\ -\frac{1} {2} \\ 1 \\ 0 \end{bmatrix} \]
\[ \lambda = 1 \longrightarrow multiplicity = 2 \]
So the eigenvalues of the given matrix will be:
\[ Eigenvalues\ =\ 1, 4, 3 \]
Numerical Result
The eigenvalues of the given matrix according to their multiplicities are:
\[ 1, 4, 3 \]
Example
Find the eigenvalues of the given matrix and list them according to their multiplicities.
\[ \begin{bmatrix} 3 & 6 & 5 \\ 0 & 2 & 0 \\ 0 & 0 & 5 \end{bmatrix} \]
As the given matrix is an upper triangular matrix, the main diagonal contain the eigenvalues. We need to check for the multiplicity of these eigenvalues as well. The multiplicities are given as:
The eigenvector of $\lambda = 3$ is given as:
\[ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \]
\[ \lambda = 3 \longrightarrow multiplicity = 1 \]
The eigenvector of $\lambda = 2$ is given as:
\[ \begin{bmatrix} -6 \\ 1 \\ 0 \end{bmatrix} \]
\[ \lambda = 2 \longrightarrow multiplicity = 1 \]
The eigenvector of $\lambda = 5$ is given as:
\[ \begin{bmatrix} 2.5 \\ 0 \\ 1 \end{bmatrix} \]
\[ \lambda = 5 \longrightarrow multiplicity = 1 \]
All the eigenvalues have the same multiplicity, we can list them in any order.
The eigenvalues of the given matrix are 3, 2, and 5.