Practice Questions on Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are special numbers and vectors associated with a matrix. They describe directions that do not change direction when a matrix transforms them—they only get stretched, compressed, or flipped.

Solved Examples

Example 1: Find the eigenvectors of the matrix A = \begin{bmatrix}1 & 1 & 0\\0 & 1 & 1\\0 & 0 & 1\end{bmatrix}

Solution:

The eigen values of the matrix is found using,
|A - λI| = 0
\begin{bmatrix}1-λ & 1 & 0\\0 & 1-λ & 1\\0 & 0 & 1-λ\end{bmatrix} = 0
(1 - λ)³ = 0

Thus, the eigen values are, λ = 1, 1, 1

As the all the eigenvalues are equal we have three identical eigenvectors. We will find the eigenvectors for λ = 1, using (A - λI)v = O

\begin{bmatrix}1-1 & 1 & 0\\0 & 1-1 & 1\\0 & 0 & 1-1\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}0\\ 0\\0\end{bmatrix}

\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix}.\begin{bmatrix}a\\ b\\c\end{bmatrix} = \begin{bmatrix}0\\ 0\\0\end{bmatrix}
solving the above equation we get,

• a = K
• y = 0
• z = 0

Then the eigenvector is,
\begin{bmatrix}a\\ b\\c\end{bmatrix}= \begin{bmatrix}k\\ 0\\0\end{bmatrix} = k\begin{bmatrix}1\\ 0\\0\end{bmatrix}

Example 2: Find the eigenvectors of the matrix A = \begin{bmatrix}5 & 0\\0 & 5 \end{bmatrix}

Solution:

The eigen values of the matrix is found using,
|A - λI| = 0
\begin{bmatrix}5-λ & 0\\0 & 5-λ \end{bmatrix} = 0
(5 - λ)² = 0

Thus, the eigen values are,
λ = 5, 5

As the all the eigenvalues are equal we have three identical eigenvectors. We will find the eigenvectors for λ = 5, using 
(A - λI)v = O
\begin{bmatrix}5-5 & 0 \\ 0 & 5-5\end{bmatrix}.\begin{bmatrix}a\\ b\end{bmatrix} = \begin{bmatrix}0\\ 0\end{bmatrix}

Simplifying the above we get,
a = 1, b = 0
a = 0, b = 1

Then the eigenvector is,
\begin{bmatrix}a\\ b\end{bmatrix}= \begin{bmatrix}1\\ 0\end{bmatrix} , \begin{bmatrix}0\\ 1\end{bmatrix}

Example 3: Given matrix A = \begin{pmatrix}4 & 1 \\2 & 3\end{pmatrix}, find the eigenvalues and eigenvectors.

Solution:

1) Find Eigenvalues:

The eigenvalues are found by solving the characteristic equation det⁡(A−λI) = 0.

A−λI= \begin{pmatrix} 4−λ & 1 \\ 2 & 3−λ \end{pmatrix}

Determinant: (4−λ)(3−λ)−2⋅1 = 0.

λ²−7λ+10 = 0.

Solving for λ: λ = 5 or λ = 2.

2) Find Eigenvectors:

For λ = 5: A−5I = \begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix}.

Solving (A−5I)x = 0 gives eigenvector x= k\begin{pmatrix} 1 \\ 1 \end{pmatrix} where k is a scalar.

For λ=2: A−2I= \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}.

Solving (A−2I)x = 0, gives eigenvector x= k\begin{pmatrix} -1 \\ 2 \end{pmatrix} where k is a scalar.

Example 4: In mechanical engineering, eigenvalues can be used to determine natural frequencies of a system. Consider a 2-DOF mass-spring system with mass matrix M= \begin{pmatrix}2 & 0 \\0 & 1\end{pmatrix} and stiffness matrix K = \begin{pmatrix}5 & -1 \\-1 & 5\end{pmatrix}

Find the natural frequencies.

Solution:

Form the generalized eigenvalue problem: det⁡(K−λM)=0

Find Eigenvalues:

K−λM = \begin{pmatrix} 5−2λ & -1 \\ -1 & 5−λ \end{pmatrix}

Determinant: (5−2λ)(5−λ) − { (-1)(−1) } = 0

2λ²−15λ +24=0

Solving for λ: λ = 5.186 or λ = 2.313

Natural Frequencies:

ω1 ≈ 2.2775, ω2 ≈ 1.5211

Practice Problems

1. Given matrix A = \begin{pmatrix}6 & 2 \\2 & 3\end{pmatrix}

• Find the eigenvalues.
• Find the corresponding eigenvectors.

2. An electrical circuit has an impedance matrix Z = \begin{pmatrix}4 & 1 \\1 & 3\end{pmatrix}. Determine the eigenvalues and eigenvectors to simplify the circuit analysis.

3. Consider a mass-spring system with mass matrix M= \begin{pmatrix}3 & 0 \\0 & 2\end{pmatrix} and stiffness matrix K= \begin{pmatrix}7 & 1 \\1 & 4\end{pmatrix} Find the natural frequencies of the system.

4. Given the covariance matrix Σ = \begin{pmatrix}20 & 80 \\81 & 5\end{pmatrix}

• Find the eigenvalues.
• Find the eigenvectors.
• Use the eigenvectors to determine the principal components.

5. A communication channel is represented by the matrix A = \begin{pmatrix}1 & 0 \\4 & 5\end{pmatrix}. Calculate the eigenvalues and eigenvectors to determine the fundamental modes of the channel.