Is a symmetric matrice allways nonsingular?

Thread starter hyper
Start date Aug 5, 2009
Tags

Symmetric

In summary, a symmetric matrix is a square matrix that is equal to its transpose, while a nonsingular matrix is a square matrix that has an inverse. However, a symmetric matrix is not always nonsingular, as it depends on the eigenvalues of the matrix. There is no direct relationship between symmetry and singularity in a matrix, but a symmetric matrix can be singular if it has at least one zero eigenvalue. To determine if a symmetric matrix is nonsingular, we can calculate its eigenvalues or check if its determinant is non-zero.

Aug 5, 2009

hyper

Can someone confirm this? If so, are there any respected websites on the net that can confirm this theorem?

Physics news on Phys.org

Aug 5, 2009

D H

Staff Emeritus

Science Advisor

Insights Author

No. Example: The zero matrix is symmetric.

Related to Is a symmetric matrice allways nonsingular?

1. What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose. This means that the elements above and below the main diagonal are reflections of each other.

2. What does it mean for a matrix to be nonsingular?

A nonsingular matrix, also known as an invertible matrix, is a square matrix that has an inverse. This means that the matrix can be multiplied by its inverse to give the identity matrix.

3. Is a symmetric matrix always nonsingular?

No, a symmetric matrix is not always nonsingular. A symmetric matrix is nonsingular if and only if all its eigenvalues are non-zero.

4. What is the relationship between symmetry and singularity in a matrix?

There is no direct relationship between symmetry and singularity in a matrix. However, a symmetric matrix can be singular if it has at least one zero eigenvalue.

5. How can we determine if a symmetric matrix is nonsingular?

To determine if a symmetric matrix is nonsingular, we can calculate its eigenvalues. If all of the eigenvalues are non-zero, then the matrix is nonsingular. Alternatively, we can also check if the determinant of the matrix is non-zero, as a nonsingular matrix will always have a non-zero determinant.