# Calculating Eigenvalue for 3D Matrix

• gaffareee
In summary, the conversation discusses finding the eigenvalue of a 3D matrix, A, which is only possible through 3D inversion or eigenvalue calculation. The speaker suggests using determinants to find the eigenvalues and provides a method for solving for the eigenvalues in a 3x3 matrix using a cubic equation.

#### gaffareee

Dear frnds,

suppose one have a 3D matrix,

A=ones(3,3,3);

he wants to have eigen value of A,
then,

A-Lamda*I=0 where A is 3D matrix, I is 3D matrix.
now

Problem is our mathematical soft can only do 2D inversion or eigen calculation.
Please refer me any info to find out the eigenvalue of 3d matrix.

thanks anyway.

I'm sorry, what do you mean by "ones(3,3,3)"? if there were only two "3"s I would interpret that as
$$\begin{bmatrix}1 & 1 & 1 \\ 1 & 1 & 1\\ 1 & 1 & 1\end{bmatrix}$$
the 3 by 3 matrix consisting of all "1"s.

Do you only know how to use software to find eigenvalues? What about determinants? I think you have found a problem with relying on software! Do you know the definition of "eigenvalue"?

One of the things you need to know is that $A- \lambda I$ does NOT have to be the 0 vector, although, if you really meant "ones(3,3)", that does happen. A number, $\lambda$, is an eigenvalue if and only if there exist a non-zero vector v such that $(A- \lambda I)v= 0$.

To find the eigenvalues of a 3 by 3 matrix, use the fact that if $(A- \lambda)v= 0$ for a non-zero vector, then the matrix $A- \lambda I$, which is just A with the variable $\lambda$ subtracted from each of its diagonal elements), while it is not necessarily 0, cannot have an inverse and so its determinant must be 0.

For the matrix above, that means you need to solve
$$\left|\begin{array}{ccc}1-\lambda & 1 & 1 \\ 1 & 1-\lambda & 1 \\ 1 & 1 & 1-\lambda\end{array}\right|= 0$$
That will give a cubic equation in $\lambda$ which might have up to three different roots. Since it should be clear that the determinant of A itself if 0, one of those roots should be obvious.

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## 1. What is an eigenvalue for a 3D matrix?

An eigenvalue for a 3D matrix is a scalar value that represents how a linear transformation will stretch or compress a vector in a particular direction.

## 2. How do you calculate eigenvalues for a 3D matrix?

To calculate eigenvalues for a 3D matrix, you first need to find the determinant of the matrix. Then, you need to solve the characteristic equation, which is a polynomial equation involving the eigenvalue as the variable. Finally, you can use algebraic methods or numerical methods to solve for the eigenvalues.

## 3. Why is calculating eigenvalues important in 3D graphics and computer vision?

Calculating eigenvalues for 3D matrices is important in computer graphics and computer vision because it allows us to understand the transformations that occur in a 3D space. This information is essential for tasks such as image processing, pattern recognition, and 3D object reconstruction.

## 4. Can a 3D matrix have multiple eigenvalues?

Yes, a 3D matrix can have multiple eigenvalues. The number of eigenvalues is equal to the dimensions of the matrix. For example, a 3x3 matrix will have three eigenvalues.

## 5. How are eigenvalues used in principal component analysis?

In principal component analysis, eigenvalues are used to determine the most significant components of a dataset. The eigenvectors associated with the largest eigenvalues represent the principal components, which can then be used to reduce the dimensionality of the dataset and extract important information.