In the world of matrix problems, **Eigenvalue **and **Eigenvector **are one of the most important terms. They are widely used to solve linear equations reducing dimension space. In this article, let us learn how to calculate **eigenvalues and eigenvectors in Excel** with **2** useful methods.

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## What Are Eigenvalues?

The **Eigenvalues **are a set of scalar values associated with a set of linear equations that has a non-zero absolute solution. People also call it **Characteristic Roots**. The process results in a set of **Determinants** that has this equation:

**(A – λI)=0**

Here, the scalar **Lambda (λ) **is the required eigenvalue that is associated with square **k x k matrix A** and **Identity matrix I**.

## What Are Eigenvectors?

In a linear equation, along with calculating **Eigenvalues**, we also calculate the **Eigenvector** of a matrix. It is a sort of vector that is generated through the scalar values. It is also called proper vectors, characteristic vectors, or latent vectors. We can symbolize it with an **X** in the **Determinant** equation which results in this:

**(A – λI)X=0**

## 2 Useful Methods to Calculate Eigenvalues and Eigenvectors in Excel

So far, we learned about **eigenvalues **and **eigenvectors **in brief. Now let us calculate them with the following **3X3 Matrix** **A**.

### 1. Calculate Eigenvalues and Eigenvectors with Goal Seek in Excel

In this first method, we will process the calculation with the **Goal Seek** feature in Excel. Let’s see how it works.

- First, insert a general
**Identity Matrix**in the**Cell range F5:H7**where we have**1**in the diagonal cells.

- Second, create a new column to find the
**Determinant**where the initial scalar**Lambda (λ)**is**0**.

- Then, insert this formula in
**Cell B11**to find the 1st value of the**k x k matrix**based on the determinant equation.

`=B5-$C$10*F5`

- Similarly, apply the same formula based on each cell of the
**A matrix**. - Therefore, the output looks like this.

- Now, insert this formula into
**Cell C15**to find the**Determinant**value.

`=MDETERM(B11:D13)`

- Next, go to the
**Data**tab and select**Goal Seek**from the**What-If Analysis**option.

- Then, refer
**Determinant**value as the**Set cell**and**Lambda (λ)**value as**By changing cell**. - Along with this, determine the
**To value**as**0**as we want the**Determinant**value equal to zero.

- Following this, press
**OK**and after some time you will get the new**Eigenvalue**along with the**Determinant**value in**Cells C10**and**C15**respectively.

- Next, change the
**Lambda (λ)**value to**1.5**and run the same process in**Goal Seek**to get another set of values as follows.

- Similarly, insert
**3**as**Lambda (λ)**value and run the**Goal Seek**operation to get the following output.

- So far, we got
**3**sets of eigenvalues as the**Lambda (λ)**value. - Lastly, insert this formula for each set of
**eigenvalues**and get the final**eigenvectors**like this.

`=-MMULT(MINVERSE(C12:D13),B6:B7)`

**the MMULT function**calculates the product of the

**k x k matrix**both the initial and the new one. Along with it,

**the MINVERSE function**provides the inverse of the matrix.

**Read More: How to Calculate Vector Multiplication in Excel (2 Useful Examples)**

### 2. Apply Power Method to Get Eigenvalues and Eigenvectors

We can apply the **Power Method** in **Excel **to find the **eigenvalues **and **eigenvectors **along with the dominant one. To do this, follow the simple steps below.

- In the beginning, create an initial vector column with the following values in the
**Cell range F5:F7**.

- Then, insert the values as a row vector in the
**Cell range C10:E10**for ease of the process.

- Next, insert this formula in
**Cell F10**to transpose and return the absolute vector value.

`=TRANSPOSE(ROUND(MMULT($B$5:$D$7,TRANSPOSE(C10:E10)),4))`

**the TRANSPOSE function**converts any given vertical range to a horizontal range and vice-versa. Then,

**the ROUND function**returns the output as a rounded number. Lastly, the

**MMULT**function calculates the product of the matrix

**$B$5:$D$7**, where we set it as an

**absolute cell reference**. Lastly, we provide

**4**in the

**num-digits**argument to get up to

**4 decimal places**.

- Afterward, press
**Enter**and you will get the following output where**k=0**.

- Next, insert this formula in
**Cell I10**to find the**Maximum Absolute Eigenvalue**.

`=MAX(ABS(F10:H10))`

Here, **the MAX function** returns the maximum value from the **Cell range** **F10:H10**. Also, **the ABS function** returns an absolute value of this omitting negative values.

- Next, select the
**Cell range C11:E11**and insert this formula to get the next horizontal vector value where**k=1**.

`=F10:H10/I10`

- Following this, select the
**Cell range F10:I10**and drag its bottom corner downward as shown below.

- After this, insert this formula in
**Cell J11**to get the absolute value of**Epsilon (e)**in the power method.

`=ABS(I11-I10)`

- Lastly, select the
**Cell range B11:J11**and drag the bottom corner of the last cell downward until it shows repetitive values like this.

- So far, we got the
**eigenvalues**(**mk+1**) and**eigenvectors**(**V[k]T**) for each value of**k**. - Therefore, analyzing the dataset above, delete the rows that have repetitive values.
- Accordingly, comparing the last value of
**Epsilon (e)**which is**0.0001**with the standard value**0.0005**we can get that**0.0001**<**0.0005**. - As a result, we have got our dominant
**eigenvalue**and**eigenvectors**based on the adjacent cell values as shown below.

**Read More: How to Apply Vector Formula in Excel (5 Suitable Examples)**

## Conclusion

Finally, we are at the end of our article. Here we tried to explain how to calculate **eigenvalues and eigenvectors** in **Excel **with **2** useful methods. Let us know your feedback in the comment box. Follow **ExcelDemy **for more tutorials.