Eigenvalue Calculator

e Calculator helps to calculate the eigenvalues of a 2 × 2 matrix. Eigenvalues are associated with eigenvectors and are used to analyze linear transformations.

What is Eigenvalue Calculator?

Eigenvalue Calculator is an online tool that helps to compute the eigenvalues for a given 2 × 2 matrix. Eigenvalues can also be defined as a special set of scalars that are associated with a system of linear equations. To use the eigenvalue calculator, enter the values in the given input boxes.

Eigenvalue Calculator

How to Use Eigenvalue Calculator?

Please follow the steps below to compute the eigenvalues of a 2 × 2 matrix using the eigenvalue calculator.

• Step 1: Go to Cuemath's online eigenvalue calculator.
• Step 2: Enter the values in the given input boxes of the eigenvalue calculator.
• Step 3: Click on the "Calculate" button to find the eigenvalues for a given 2 x 2 matrix.
• Step 4: Click on the "Reset" button to clear the fields and enter new values.

How Does Eigenvalue Calculator Work?

Suppose we have a square matrix given by \(A_{n\times n}\). Let λ be a scalar quantity. Then [A - λI] is known as an Eigen or a characteristic matrix. Here, I is used to represent the identity matrix. The determinant of this characteristic matrix can be given by | A - λI | and the Eigen equation will be | A - λI | = 0. To find the Eigenvalues of this equation the following steps are utilized.

• Let A be a 2 × 2 square matrix.
• The identity matrix I = \(\begin{bmatrix} 1 & 0\\ 0& 1 \\\end{bmatrix}\)
• Now we multiply the identity matrix I with some scalar λ. This gives us λI
• Next, we subtract λI from the matrix A; A - λI
• We then find the determinant of the matrix obtained. That is | A - λI |.
• This results in a quadratic expression.
• We equate this expression to zero. Thus, | A - λI | = 0.
• Finally, we solve the quadratic equation to get two values of λ. These two values will be the eigenvalues.

Solved Examples on Eigenvalue Calculator

Example 1:

Find the eigenvalue of \(\begin{bmatrix} 0 & 1\\ 2& 3 \\\end{bmatrix}\) and verify it using the eigenvalue calculator.

Solution:

Given matrix: A = \(\begin{bmatrix} 0 & 1\\ 2& 3 \\\end{bmatrix}\)

| A - λI | = 0, where I is identity matrix i.e.,

A - λI = \(\begin{bmatrix} 0 & 1\\ 2& 3 \\\end{bmatrix}\) - λ\(\begin{bmatrix} 1 & 0\\ 0& 1 \\\end{bmatrix}\)

A - λI = \(\begin{bmatrix} - λ & 1\\ 2 & 3 - λ \\\end{bmatrix}\)

| A - λI | = λ² - 3λ - 2

Substituting these values in | A - λI | = 0 we get

λ² - 3λ - 2 = 0

On solving,

λ = -0.56 , 3.56

Example 2:

Find the eigenvalue of \(\begin{bmatrix} 1.2 & 3.4\\ 1.6& 3 \\\end{bmatrix}\) and verify it using the eigenvalue calculator.

Solution:

Given matrix: A = \(\begin{bmatrix} 1.2 & 3.4\\ 1.6& 3 \\\end{bmatrix}\)

| A - λI | = 0, where I is identity matrix i.e.,

A - λI = \(\begin{bmatrix} 1.2 & 3.4\\ 1.6& 3 \\\end{bmatrix}\) - λ\(\begin{bmatrix} 1 & 0\\ 0& 1 \\\end{bmatrix}\)

A - λI = \(\begin{bmatrix} 1.2 - λ & 3.4\\ 1.6 & 3 - λ \\\end{bmatrix}\)

| A - λI | = λ² - 4.2λ - 1.84

Substituting these values in | A - λI | = 0 we get

λ² - 4.2λ - 1.84 = 0

On solving,

λ = -0.4 , 4.6

Similarly, you can try the eigenvalue calculator to find the value of the eigenvalues for the following matrices:

• \(\begin{bmatrix} 8 & 9\\ 10& 5.6 \\\end{bmatrix}\)
• \(\begin{bmatrix} 0.25 & 1.3\\ 7.8& 0.5 \\\end{bmatrix}\)

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• Matrices
• Eigenvalues

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