Eigenvectors of a symmetric matrix.

In summary, for an nxn symmetric matrix, it is true that there will be n linearly independent eigenvectors even if the eigenvalues are not distinct. To prove this rigorously, it can be shown that if an nxn symmetric matrix has an eigenvalue of 0 with multiplicity k, then its rank will be n-k. This can be proven by finding k linearly independent eigenvectors that solve Ax=0 and using the rank-nullity theorem. According to the link provided, symmetric matrices are diagonalizable, which also supports this result.
  • #1
chocolatefrog
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Is it true that an nxn symmetric matrix has n linearly independent eigenvectors even for non-distinct eigenvalues? How can we show it rigorously? Basically, I want to prove that if an nxn symmetric matrix has eigenvalue 0 with multiplicity k, then its rank is (n - k). If we can prove that there exist k linearly independent eigenvectors which solve Ax = 0, then we can use the rank-nullity theorem to directly show the result, right?
 
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FAQ: Eigenvectors of a symmetric matrix.

What are eigenvectors of a symmetric matrix?

Eigenvectors of a symmetric matrix are special vectors that, when multiplied by the matrix, result in a scalar multiple of the original vector. They represent the directions along which the matrix only stretches or compresses, without changing its orientation.

How do you find eigenvectors of a symmetric matrix?

To find the eigenvectors of a symmetric matrix, you first need to find its eigenvalues. This can be done by solving the characteristic equation det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. Once you have the eigenvalues, you can find the corresponding eigenvectors by solving the system of linear equations (A - λI)v = 0, where v is the eigenvector.

Why are eigenvectors of a symmetric matrix important?

Eigenvectors of a symmetric matrix are important because they are used in many applications, such as data analysis, image processing, and quantum mechanics. They also have special properties that make them useful for solving systems of linear equations and understanding the behavior of a matrix.

Can a symmetric matrix have complex eigenvectors?

Yes, a symmetric matrix can have complex eigenvectors. In fact, symmetric matrices always have a complete set of orthogonal eigenvectors, which can be real or complex. The complex eigenvectors can be converted into real eigenvectors by taking the real and imaginary parts separately.

How are eigenvectors of a symmetric matrix related to its diagonalization?

Eigenvectors of a symmetric matrix are used to diagonalize the matrix, which means to transform it into a diagonal matrix. This is done by finding a matrix P, where the columns are the eigenvectors of the matrix, and then computing P-1AP. This diagonalized matrix will have the eigenvalues along the diagonal, and the corresponding eigenvectors will form the columns of P.

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