- #1
gamerninja213
- 3
- 0
Homework Statement
If the only eigenvalue is zero, can you ever get a set of n linearly independent vectors?
A diagonal matrix is a square matrix where all the elements outside the main diagonal (from top left to bottom right) are zero. This means that all the non-zero elements are located on the main diagonal.
Eigenvalues are a set of numbers associated with a square matrix that indicate the scaling factor of the corresponding eigenvectors. In other words, they represent the amount by which a vector is stretched or compressed when multiplied by the matrix.
Yes, it is possible for a diagonal matrix to have all eigenvalues equal to zero. This means that all the vectors in the matrix are scaled by a factor of zero when multiplied by the matrix, resulting in no change.
Diagonal matrices with all eigenvalues equal to zero are used in various mathematical operations, such as finding the inverse of a matrix and solving systems of linear equations. They also have applications in fields like physics and engineering where they are used to represent physical quantities.
Identity matrices have all eigenvalues equal to one, while diagonal matrices with all eigenvalues equal to zero have all their non-diagonal elements equal to zero. This means that identity matrices do not change the magnitude of a vector when multiplied, while diagonal matrices with all eigenvalues equal to zero result in a scaled down or zero vector.