\[ \begin{bmatrix} 4 & -5 & 7 & 0 \\ 0 & 3 & 1 & -5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

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.**