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.
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What Are Eigenvectors?
A square matrix is a must requirement for Eigenvector. It is a non-vector component that we get from the scalar multiplication of the given matrix. The equation related to the eigenvector is:
A= Given a matrix of size n x n v=Eigenvector λ=Eigenvalue
A= Given a matrix of size n x n
It is called the Eigenvector equation.
The eigenvector is also known as the characteristic vector, and the eigenvalue is known as the characteristic value.
Read More: How to Calculate Vector Multiplication in Excel (2 Useful Examples)
Steps to Calculate Eigenvectors in Excel
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 being 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.
📌 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.
📌 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 λ1. 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.
- We need to copy the 1st and 2nd column of matrix A-λI on Cell E15 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 will use the MDETERM function to calculate the eigenvectors. Look at the formulas used on Cells G15:G17.
Formula in Cell G15:
In Cell G16:
In Cell G17:
- Now, optimize the eigenvector and set that on Cell L7 using the following formula.
- 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.
- 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. And all eigenvectors are column vectors.
Read More: How to Calculate Eigenvalues and Eigenvectors in Excel
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 have a look at our website ExcelDemy and give your suggestions in the comment box.