Finding Eigenvectors by inspection

• Nothing000
In summary, by inspecting the elements of a simple symmetric matrix, one can identify eigenvectors by observing when the lines of the matrix sum to the same value. This can be used to determine the associated eigenvalue, and the number of eigenvalues in a matrix is at most equal to the size of the matrix. To find the eigenvalues, one must solve the characteristic equation and the eigenvectors are the non-zero vectors in the eigenspace associated with each eigenvalue. The eigenspace can be found by taking the kernel of the matrix subtracted by the identity matrix multiplied by the eigenvalue.
Nothing000
Would someone please explain to me how I can find eigenvalues and eigenvectors by inpection of simple symmetric matrices? I just can't figure it out.

He is an example:
By looking at $$A=\left(\begin{matrix}2&-1&-1\\-1&2&-1\\-1&-1&2\end{matrix}\right)$$ I should be able to guess that [1,1,1] is an eigenvector.

Last edited:
Because the elements in the lines sum to the same thing. when you multiply the matrix by the vector (1,1,1), you get a new vector whose components are the sum of each lines in the matrix. as soon as the lines of a matrix sum to the same thing, then (1,1,1) is an eigenvector with associated eigenvalue the sum of the lines, in this case 2-1-1=0.

How do I know how many eigenvalues there will be. In other words, if I could see that [1,1,1] is an eigenvector just by inspection, then how do I know if there are any other eigenvectors?

The eigenvectors are spanned by [1,1,1]. An example would be [2,2,2] is an eigevector of A. Also, an nxn matrix has at most n eigenvalues (counted with their algebraic multiplicities). To find the eigenvalues you just have to solve the characteristic equation. Then the eigenvectors are the nonzerovectors in the eigenspace associated with that eigenvalue. Eigenspace with eigenvalue L is ker(A-LI) where I is the nxn identity

1. What is the concept of "Finding Eigenvectors by inspection"?

"Finding Eigenvectors by inspection" is a method used in linear algebra to quickly and easily find the eigenvectors of a given matrix. It involves observing patterns in the matrix and using algebraic manipulations to determine the eigenvectors without having to perform any complex calculations.

2. How is "Finding Eigenvectors by inspection" different from other methods of finding eigenvectors?

Unlike other methods, "Finding Eigenvectors by inspection" does not require us to solve for the eigenvalues of a matrix first. Instead, we use the properties of eigenvectors and eigenvalues to identify patterns in the matrix that can help us determine the eigenvectors.

3. Can "Finding Eigenvectors by inspection" be applied to any type of matrix?

Yes, the method can be applied to any square matrix, regardless of its size or elements. However, it may not always be possible to find all the eigenvectors by inspection, and in such cases, other methods may be necessary.

4. How accurate is the "Finding Eigenvectors by inspection" method?

The accuracy of this method depends on the ability to identify patterns in the matrix and perform algebraic manipulations correctly. If done accurately, the method will provide the correct eigenvectors of the matrix.

5. Is it necessary to have prior knowledge of linear algebra to use "Finding Eigenvectors by inspection"?

While having some knowledge of linear algebra can be helpful, it is not necessary to use this method. With practice and understanding of the properties of eigenvectors and eigenvalues, anyone can successfully find eigenvectors by inspection.

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