Eigenvalues and Eigenvectors (Examples & Questions)

Eigenvalues and eigenvectors help solve varied problems. What are they, how do we find and use them?

The vectors that are scaled by matrix multiplication are called eigenvectors of a matrix. The scale factor is called the eigenvalue.

We can multiply a square matrix with a vector to get a new vector. We say the matrix has transformed the old vector into a new vector. Usually, the new vector has a different direction, except for a few special ones. These are the eigenvectors. They get scaled by a factor called the eigenvalue. If the scale factor is negative, it reverses the direction of the eigenvector.

Finding Transformation Matrix

The matrix $A=\left(\begin{array}{cc}1& 2\\ 3& 0\end{array}\right)$ rotates, flips and stretches the entire space ( ℝ ² ). Only eigenvectors preserve their direction. All other vectors point to a new direction.

Let us do the transformation ourselves. We start with the unit vectors.

Multiplying A with the unit vector i = (1, 0)^T:

$Ai=\left(\begin{array}{cc}1& 2\\ 3& 0\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}1\\ 3\end{array}\right)$.

You can verify the transformation of the unit vector j as:

(0, 1)^T → $\left(\begin{array}{c}2\\ 0\end{array}\right)$

Observe that transformed unit vectors and columns of the matrix are identical.

Eigenvector Transformation

Let us transform vectors v₁ = ( − 1, 1.5)^T and v₂ = (1, 1)^T.

$A{v}_1=A\left(\begin{array}{c}-1\\ 1.5\end{array}\right)=\left(\begin{array}{c}2\\ -3\end{array}\right)$

= − 2( − 1, 1.5)^T = − 2v₁

It is scaled by a factor of −2. Due to the negative sign of the scale factor, its direction gets reversed.

v₁ is an eigenvector. The eigenvalue corresponding to this vector is −2.

$A{v}_2=A\left(\begin{array}{c}1\\ 1\end{array}\right)$

$=\left(\begin{array}{c}3\\ 3\end{array}\right)=3\left(\begin{array}{c}1\\ 1\end{array}\right)=3v_2$

v₂ is also an eigenvector. It is scaled by a factor of 3. Its eigenvalue is 3.

Finding Eigenvalue

The eigenvalue is the amount by which a square matrix scales its eigenvector.

If x is an eigenvector of a matrix A, and λ its eigenvalue, we can write:

Ax = λx where A is an n × n matrix.

We want to solve this equation for λ and x( ≠ 0). Rewriting the equation:

Ax − λx = 0

(A − λI)x = 0

This multiplication will result in n linear equations. Since there is no constant term, this is a homogeneous system of equations (you will see this in the examples below). It has a solution if and only if:

Determinant of (A − λI) = 0

On calculating the determinant you get a polynomial p(λ) of degree n. So at most n possible roots exist or a maximum of n different eigenvalues exist for an n × n matrix.

Let us see an example.

Example

Find eigenvalues of the 2 x 2 matrix $A=\left(\begin{array}{cc}1& 2\\ 3& 0\end{array}\right)$.

We use the following equation to calculate eigenvalues:

Determinant of (A − λI) = 0

Step 1. Form Matrix

$A-\lambda I=\left(\begin{array}{cc}1-\lambda & 2\\ 3& 0-\lambda \end{array}\right)$

Step 2. Create Equation

det(A − λI) = 0 gives the equation:

(1 − λ)( − λ) − 6 = 0

The 2 × 2 matrix, resulted in a second-degree polynomial in our equation.

λ² − λ − 6 = 0

Step 3. Solve

(λ − 3)(λ + 2) = 0

λ = 3 or λ = − 2

Eigenvalues are 3 and -2

Finding Eigenvector

Eigenvector or characteristic vector of a matrix is a unique vector that does not change its direction (except reversal) under matrix multiplication.

• Matrix multiplication transforms any vector.
• w = Av is the transformed vector v. Generally v and w point in different directions.
• v and w are in the same direction if v is an eigenvector.

Example

Find eigenvectors of $A=\left(\begin{array}{cc}1& 2\\ 3& 0\end{array}\right)$.

Step 1. Find Eigenvalues

We have already calculated the eigenvalues as -2 and 3. First we find the eigenvector for λ = − 2.

Step 2. Form Equations for λ = − 2

(A − Iλ) =

$\left(\begin{array}{cc}1-(-2)& 2\\ 3& 0-(-2)\end{array}\right)=\left(\begin{array}{cc}3& 2\\ 3& 2\end{array}\right)$

Step 3. Solve Equations

We now solve (A − Iλ)x = 0

$\left(\begin{array}{cc}3& 2\\ 3& 2\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=0$

We get the equation:

3x₁ + 2x₂ = 0

$x_1=\frac{-2}{3}{x}_2$

We can choose any value for x₂ to calculate the eigenvector.

For x₂ = 3 we get x₁ = − 2

Eigenvector is ( − 2, 3)^T

Any value of x₂ results in an eigenvector. They are all scaled by a factor. Any scalar multiple of an eigenvector is also an eigenvector.

Eigenvector for λ = 3:

Again we calculate (A − Iλ) =

$\left(\begin{array}{cc}1-3& 2\\ 3& 0-3\end{array}\right)=\left(\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right)$

We now solve (A − Iλ)x = 0

$\left(\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=0$

We get the equation:

−2x₁ + 2x₂ = 0

x₁ = x₂

You may verify that the other equation is the same.

Again, we can choose any value for x₂ to calculate an eigenvector.

For x₂ = 1 we get x₁ = 1

Eigenvector is (1, 1)^T

Example

Find eigenvalues and eigenvectors corresponding to counterclockwise rotation through the angle π/2 about the origin in ℝ ².

Finding Rotation Matrix

Where do the unit vectors get mapped? Counterclockwise rotation will transform them as:

• $i=\left(\begin{array}{c}1\\ 0\end{array}\right)\to \left(\begin{array}{c}0\\ 1\end{array}\right)$. This is the first column of our matrix.
• $j=\left(\begin{array}{c}0\\ 1\end{array}\right)\to \left(\begin{array}{c}-1\\ 0\end{array}\right)$. This is the second column of our matrix.

Our rotation matrix is$A=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$

Calculating Eigenvalues

$A-\lambda I=\left(\begin{array}{cc}-\lambda & -1\\ 1& -\lambda \end{array}\right)$

det(A − λI) = 0 gives the equation:

λ² + 1 = 0

$\lambda =\pm \sqrt{-1}$

λ = i or λ = − i

The matrix was real, but eigenvalues are complex!

Calculating Eigenvectors

Solving for eigenvectors of λ = i:

$\left(\begin{array}{cc}0-i& -1\\ 1& 0-i\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=0$

We get the following result:

−ix₁ − x₂ = 0

−ix₁ = x₂

x₁ = ix₂

For x₂ = 1 we get x₁ = i

Eigenvector is (i, 1)^T

Solving for eigenvectors of λ = − i:

$\left(\begin{array}{cc}0+i& -1\\ 1& 0+i\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=0$

We get the following result:

ix₁ − x₂ = 0

$x_1=\frac{x_2}{i}$

x₁ = − ix₂

For x₂ = 1 we get x₁ = − i

Eigenvector is ( − i, 1)^T

There are no real eigenvectors. This transformation rotates all the points in ℝ ² (except origin). No vector preserves its direction. The absence of a real eigenvector is expected.

Properties

Eigenvectors (and eigenvalues) have interesting properties.

Ax = λx, where λ is an eigenvalue and x is an eigenvector of the matrix A.

1. Any scalar multiple of an eigenvector is also an eigenvector.
2. Real matrices can have complex eigenvalues and eigenvectors.
3. Complex eigenvectors (and eigenvalues) appear in conjugate pairs.
4. Matrix A and A + I have the same eigenvectors. Eigenvalues of (A + I) = λ + 1.
5. Matrix A and A⁻¹ have the same eigenvectors. Eigenvalues of A⁻¹ = $\frac{1}{\lambda }$.
6. Matrix A and A² have the same eigenvectors. Eigenvalues of A² = λ².
7. Eigenvalue of the identity matrix is 1

Example

What is the relationship between the eigenvalues of a matrix and its inverse? What about eigenvectors?

Suppose eigenvalue of the matrix A⁻¹ = β. Let the corresponding eigenvector be w.

We can write:

A⁻¹w = βw

AA⁻¹w = βAw

Iw = w = βAw

$\frac{1}{\beta }w=Aw$

$Aw=\frac{1}{\beta }w$

So $\frac{1}{\beta }$ is the eigenvalue of matrix A and w is its eigenvector.

A matrix and its inverse have the same eigenvectors. Their eigenvalues are the multiplicative inverse of each other.

Questions

Apart from the questions below, you can attempt Problem Set 6.1 on this page at the MIT site.

1 Question

Find the eigenvalues and the eigenvectors of:

$A=\left(\begin{array}{cc}3& 7\\ 9& 1\end{array}\right)$

Answer

2 Question

The matrix $M=\left(\begin{array}{cc}-1& 3\\ 2& 0\end{array}\right)$ has eigenvalues -3 and 2.

For -3 the eigenvector is ( − 3, 2)^T and for 2 it is (1, 1)^T.

Find the eigenvalues and eigenvectors of M². What is the relation between eigenvalues and eigenvectors of M and M²?

Answer

3 Question

The following matrix represents a rotation of a drone in three-dimensional space. Find its axis of rotation.

$A=\left(\begin{array}{ccc}\frac{\sqrt{3}}{2}& 0& 0.5\\ 0& 1& 0\\ -0.5& 0& \frac{\sqrt{3}}{2}\end{array}\right)$

Answer

4 Question

Given the matrices $A=\left(\begin{array}{cc}0& 2\\ 1& 1\end{array}\right)$ and $A^{-1}=\left(\begin{array}{cc}-0.5& 1\\ 0.5& 0\end{array}\right)$, find their eigenvalues and the eigenvectors. How are the trace and the eigenvalues of a matrix related?

What is the relationship between the eigenvalues of A and A⁻¹? What about the eigenvectors?

Answer

5 Question

A matrix represents a pure rotation in three-dimensional space. Can you guess one of its eigenvalues?

Answer

Answers

1 Answer

2 Answer

3 Answer

4 Answer

5 Answer