## 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. They are also called *Characteristic Roots*. The process results in a set of *Determinants* that has this equation:

**(A â€“ Î»I)=0**

The scalar **Lambda (Î») **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 vector generated through the scalar values. We can symbolize it with an **X** in the *Determinant* equation which results in this:

**(A â€“ Î»I)X=0**

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

Let’s calculate them with the following **3X3 Matrix**Â **A**.

### Method 1 – Calculate Eigenvalues and Eigenvectors with Goal Seek in Excel

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

- Create a new column to find the
**Determinant**where the initial scalar**Lambda (Î»)**is**0**.

- Insert this formula in
**Cell B11**to find the first value of the**k x k matrix**based on the determinant equation.

`=B5-$C$10*F5`

- Apply the same formula based on each cell of the
**A matrix**. - The output looks like this.

- Insert this formula into
**Cell C15**to find the**Determinant**value.

`=MDETERM(B11:D13)`

**MDETERM**function returns the value of the determinant from the

**B11:D13**array.

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

- Refer to the
**Determinant**value as the**Set cell**and the**Lambda (Î»)**value in**By changing cell**. - Put the
**To value**as**0**as we want the**Determinant**value equal to zero.

- Press
**OK**and you will get the new**Eigenvalue**along with the**Determinant**value in**Cells C10**and**C15**respectively.

- Change the
**Lambda (Î»)**value to**1.5**and run the same process in**Goal Seek**to get another set of values as follows.

- Insert
**3**as**Lambda (Î»)**value and run the**Goal Seek**operation to get the following output.

- We got
**3**sets of eigenvalues as the**Lambda (Î»)**value. - 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.

### Method 2 – Apply the Power Method to Get Eigenvalues and Eigenvectors

- Create an initial vector column with the following values in the
**Cell range F5:F7**.

- Insert the values as a row vector in the
**Cell range C10:E10**.

- 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*.

- Press
**Enter**and you will get the following output where**k=0**.

- 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.

- Select the cell
**range C11:E11**and insert this formula to get the next horizontal vector value where**k=1**.

`=F10:H10/I10`

- Select the
**Cell range F10:I10**and drag its bottom corner downward.

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

`=ABS(I11-I10)`

- Select the
**Cell range B11:J11**and drag the bottom corner of the last cell downward.

- We got the
**eigenvalues**(**mk+1**) and**eigenvectors**(**V[k]T**) for each value of**k**. - Delete the rows that have repetitive values.
- Compare the last value of
**Epsilon (e)**which is**0.0001**with the standard value**0.0005**Â to confirm that**0.0001**<**0.0005**. - We got the dominant
**eigenvalue**and**eigenvectors**based on the adjacent cell values as shown below.

**Download the Practice Workbook**

**<< Go Back to | Vectors in Excel |Â Excel for Math | Learn Excel**

Hai, how do we know the standard value of Epsilon (e) = 0.0005? does the epsilon value is decided by us or does 0.0005 is the standard value?

Hi

HANI,Thanks for your query. The choice of epsilon depends on the desired accuracy of the eigenvalue estimate. It is typically a small positive value, and the algorithm terminates when the absolute difference between consecutive eigenvalue estimates falls below epsilon.

The specific value of epsilon will vary depending on the application and the desired level of accuracy. Using epsilon = 0.0005 indicates that the power method iterations will continue until the absolute difference between consecutive eigenvalue estimates falls below 0.0005. This level of tolerance can be suitable for many applications, especially when the eigenvalues of the matrix are relatively close in magnitude.

However, it’s important to note that the appropriate value of epsilon may vary depending on the specific problem and matrix being analyzed. If you find that the algorithm converges too quickly or does not reach the desired accuracy with epsilon = 0.0005, you may need to adjust the value accordingly.

Consider the scale and conditioning of the problem as well. If the matrix is ill-conditioned or has eigenvalues with large differences in magnitude, a smaller epsilon may be necessary to obtain accurate results.

So, it’s always important to assess the specific requirements of your problem and adjust the value of epsilon accordingly to achieve the desired accuracy.

Regards

Rafiul HasanTeam

ExcelDemyHi.Thx for your doc, I can try SVD in excel without programed library.

But I wonder, why do we must use “Round” function?

I tried to use without “round”, AV^kt was not calculated collectly.

Can you explain for me please?

Hi agian,

I solve the problem about “round” function, which is my just mistake…..

Now I wondering about ‘how can we find other Eigen Vector and Value.

In my case, I mus find eigen vectors of 10×10 matrix….

so, I practice with 3×3 matrix and try to find normalized method.

within 3×3 matrix, [5,4,-1; 4,5,1; -1,1,2],

starting with V = 0,1,0 give the eigen value = 9 but

starting with V = 0,0,1 give the eigen value = 3. how can we choose V?

Furthermore, in hand writing, eigen value = 0 is the another solution.

and I cannot find ‘V’ such as give back eigen value = 0

please help me…..

Dear MATHMAN,

We are glad that you have already got your answer related to the

ROUNDfunction.Now let’s jump to the problem related to finding other EigenValue and EigenVector using the power method.

In the power method, it is a must to choose an initial vector

V. So, the tips for what to keep in mind while randomly choosing an initial vector_{0}V:_{0}If initial vector

Vand_{0}=[0,1,1]Wis the given dominant Eigenvector, thenV_{0}X W =0ÂThe dominant Eigenvalue for the 3×3 matrix [5,4,-1; 4,5,1; -1,1,2] is approximately

Î» â‰ˆ 9.If you are getting Eignevalue

Î»â‰ˆ0for the initial EigenVectorV= [0,0,1] most probably because the vector has no magnitude along the x and y axes and a magnitude of 1 along the z-axis. Therefore, the initial vector_{0}Vdoes not converge to the dominant Eigenvalue for the given matrix. It is expected that different EigenVectors can lead to different EigenValues._{0}The dominant eigenvalue is typically the one with the largest magnitude. It is recommended to use multiple initial vectors in the power method so that you know which eigenvalue converges to the dominant eigenvalue.

I hope this solution resolves your issue. Feel free to email us at

if you have any further problems or inquiries.[email protected]ÂRegards,

Qayem Ishrak KhanTeam

ExcelDemy.