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:

`Av=λv`

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

**v**=Eigenvector

**λ**=Eigenvalue

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:

**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 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:

- 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**:

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

**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**:

`=MDETERM(F10:H12)`

**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.

**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**.

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**λ****1**. - 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**:

`=F10:H12`

- Copy the
**1st**and**2nd**column of matrix**A-λI**in**Cell E15**by 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 use the **MDETERM **function to calculate the eigenvectors.

- Insert the following formulas in
**Cells G15:G17**:

**In Cell G15:**

`=MDETERM(C15:D16)`

**In Cell G16:**

`=MDETERM(D15:E16)`

**In Cell G17:**

`=MDETERM(E15:F16)`

- Optimize the eigenvector in
**Cell L7**by using the following formula:

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

- 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:

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

- 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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