Eigenvector is related to linear algebra. When a set of equations forms a square matrix, then we will be able to form eigenvectors of that matrix. Eigenvector is very useful in differential equations and their associated applications. In this article, we will discuss how to form eigenvectors in Excel in detail.

**Table of Contents**Expand

**What Are Eigenvectors?**

A square matrix is a must requirement for Eigenvectors. It is a non-vector component that we get from the scalar multiplication of the given matrix. The equation related to the eigenvector is:

`Av=λv`

**A**= Given a matrix of size **n** x **n**

**v**=Eigenvector

**λ**=Eigenvalue

It is called the **Eigenvector** equation.

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

**How to Calculate Eigenvectors in Excel: with Detailed ****Steps**

The calculation of an eigenvector is a complex and time-consuming process mathematically. But here we need to do this in Excel. We will explain mathematical steps in this article and apply the vector formula of Excel to solve the mathematical steps. Look at the below section for details.

**📌**** Step 1: Insert Identity Matrix and Other Supporting Elements**

First, we need to calculate the eigenvalues before going for eigenvectors. We already showed the mathematical expression eigenvector, that is,

**Av=λv**

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 scaler **λ**. So, we multiplied **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)** considered to be **0**.

- We will consider the following matrix.

- We insert an
**Identity Matrix, I**of size**3×3**in the dataset. - Also, added other supporting components to the dataset.

- Now, find out
**A-λI**by putting the formula on**Cell F10**.

`=B5:D7-I4*B10:D12`

**📌 Step 2: Calculate the Determinant Using Excel MDETERM Function**

Excel has a default function, which is named **MDETERM,** for calculating determinants.

- Now, we will find out the
**det(A-λI)**using the following formula on**Cell H5**.

`=MDETERM(F10:H12)`

**📌 Step 3: Enable Excel Goal Seek Feature**

Now, we will apply the Goal Seek feature to determine the eigenvalues.

- Go to the
**Data**tab, **What-If-Analysis**option from the**Forecast**group.- Then, choose the
**Goal Seek**option.

**📌 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.

- We add new cells in the dataset for 3 eigenvalues and eigenvectors.

The **Goal Seek** window appears by following **Step 3**. We will insert cell references for two sections and set the determinant to **zero **(**0**).

- Look at the following image to set the cell.
- Finally, click on the
**OK**button.

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

- After processing, we see the optimized eigenvalue.

This is the **1st** eigenvalue.

- Now, copy
**Cell I4**by pressing**Ctrl+C**. - We will paste this value on
**Cell L5**as**λ****1**. Go to 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 will find out the corresponding eigenvector of the calculated eigenvector.

- Now, copy matrix
**A-λI**using the following equation on**Cell B15**.

`=F10:H12`

- We need to copy the
**1st**and**2nd**column of matrix**A-λI**on**Cell E15**using the following formula.

`=F10:G12`

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 will use the
**MDETERM**function to calculate the eigenvectors. Look at the formulas used on**Cells G15:G17**.

**Formula in Cell G15:**

`=MDETERM(C15:D16)`

**In Cell G16:**

`=MDETERM(D15:E16)`

**In Cell G17:**

`=MDETERM(E15:F16)`

- Now, optimize the eigenvector and set that on
**Cell L7**using the following formula.

`=G15/MIN(ABS($G$15:$G$17))`

- Now, use the paste-special technique to keep the values only.

**📌 Step 6: Calculate Rest of the Eigenvalues and Eigenvectors in Excel**

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

- Previously, 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.

`=G15/MIN(ABS($G$15:$G$17))`

- Now, set the eigenvalue to
**0**and apply the**Goal Seek**feature again.

- Finally, we get three eigenvalues and their corresponding eigenvectors.

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

**Download Practice Workbook**

Download this practice workbook to exercise while you are reading this article.

**Conclusion**

In this article, we described the eigenvalues and eigenvectors. We showed the calculation of eigenvalues and eigenvalues in Excel in detail. I hope this will satisfy your needs. Please give your feedback and suggestions in the comment box.

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