How to Calculate Eigenvectors in Excel (with Detailed Steps)

In linear algebra, when a set of equations forms a square matrix, we are able to form eigenvectors of that matrix. An eigenvector is a non-vector component that we derive from the scalar multiplication of the given square matrix.

The Eigenvector equation is:


A= Given a matrix of size n x n



The eigenvector is also known as the characteristic vector, and the eigenvalue is known as the characteristic value.

The calculation of an eigenvector is a complex and time-consuming process mathematically. In this article, we will explain the mathematical steps and apply the vector formula in Excel to solve them.

Step 1 – Insert Identity Matrix and Other Supporting Elements

First, we need to calculate the eigenvalues before we can calculate the eigenvectors. As detailed above, the mathematical expression of the eigenvector is:


or, Av-λv=0

or, v(A-λ)=0

Finally, v(A-λI)=0

As the A is a matrix, we need another matrix with the scalar λ. So we multiply λ by the Identity matrix. The Identity matrix will not change the value of the equation. In the calculation of eigenvalues, the determinant, det(A-λI) is considered to be 0.

Consider the following matrix:

Sample matrix for Eigenvectors

  • Insert an Identity Matrix, I of size 3×3 in the dataset.
  • Add other supporting components to the dataset.

  • To find A-λI, enter this formula in Cell F10:

Matrix subtraction for calculating Eigenvectors

Step 2 – Calculate the Determinant Using the MDETERM Function

Excel has a default function, MDETERM, for calculating determinants.

  • To find the det(A-λI), enter the following formula in Cell H5:

Calculate determinant of matrix for Eigenvectors

Step 3 – Enable Excel Goal Seek Feature

Now we can apply the Goal Seek feature to determine the eigenvalues.

  • Go to the Data tab,
  • Select What-If-Analysis from the Forecast group.
  • Select the Goal Seek option.

Enable goal seek feature in Excel

Step 4 – Calculate Eigenvalues

As the given matrix is 3×3, we will get 3 eigenvalues. And for each eigenvalue, we will get one eigenvector.

  • Add new cells in the dataset for 3 eigenvalues and eigenvectors.

The Goal Seek window appears by following Step 3.

  • Insert cell references for the two sections and set the determinant to zero (0) as in the following image.
  • Click OK.

Calculate Eigenvalue for Eigenvectors

Here, the eigenvalue is the changing cell that we get by using the Goal Seek feature.

After processing, we see the optimized eigenvalue.

This is the 1st eigenvalue.

  • Copy Cell I4 by pressing Ctrl+C.
  • Paste this value in Cell L5 as λ1.
  • Select the Values option of the Paste Special section.

Step 5 – Find Eigenvectors for Each of the Eigenvalues

We already calculated the 1st eigenvalue. Now we can find the corresponding eigenvector.

  • Copy matrix A-λI by using the following equation in Cell B15:

  • Copy the 1st and 2nd column of matrix A-λI in Cell E15 by using the following formula:

We get a 3×5 matrix. This will be used in the calculation of the eigenvectors. Each eigenvector must be a column matrix. As the given matrix is of size 3×3, the eigenvectors will be of size 3×1.

We use the MDETERM function to calculate the eigenvectors.

  • Insert the following formulas in Cells G15:G17:

In Cell G15:


In Cell G16:


In Cell G17:


Calculate Eigenvectors from eigenvalue

  • Optimize the eigenvector in Cell L7 by using the following formula:

Calculate Eigenvectors from eigenvalue

  • Use the paste-special technique described above to keep the values only.

Step 6 – Calculate the Rest of the Eigenvalues and Eigenvectors

We will repeat Steps 4 and 5 again to get the rest of the eigenvalues and their corresponding eigenvectors.

  • Previously, we set the value of λ as negative. Now, set this a positive value and apply the Goal Seek feature.

We get a new eigenvalue and eigenvector for that eigenvalue as shown before:


  • Set the eigenvalue to 0 and apply the Goal Seek feature again.

We get three eigenvalues and their corresponding eigenvectors.

Ultimately we get V1, V2, and V3 as the 3 eigenvectors of three eigenvalues -6,15,3 respectively. All eigenvectors are column vectors.

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Alok Paul
Alok Paul

Alok Paul has completed his B.Sc. in Electronics and Telecommunication Engineering from East West University. He has been working on the ExcelDemy project for more than 2 years. He has written 220+ articles and replied to numerous comments. He is experienced in Microsoft Office, especially in Excel. He also led some teams on Excel and VBA content development. He has a keen interest in Advanced Excel, Data analysis, Excel Pivot Table, Charts, and Dashboard. He loves to research... Read Full Bio

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