An eigenvector is defined to be a nonzero vector x such that (A - λI)x = 0. The eigenspace is the space spanned by all of the eigenvectors. Since no eigenvector can be the zero vector, the eigenspace can't be just the space with 0 in it.Hernaner28 said:Hi. I've got a theoretical doubt: why is the geometric multiplicity more or equal to 1?
Couldn't happen that the eigenspace is the null vector?
The geometric multiplicity is a measure of the number of eigenvectors corresponding to a particular eigenvalue of a matrix. Since every matrix has at least one eigenvector, the geometric multiplicity is always greater than or equal to 1.
The geometric multiplicity represents the dimension of the eigenspace associated with a specific eigenvalue. In other words, it tells us how many linearly independent eigenvectors exist for that particular eigenvalue.
No, the geometric multiplicity cannot be equal to 0. This is because every square matrix has at least one eigenvalue, and therefore at least one eigenvector. If the geometric multiplicity were 0, it would mean that there are no eigenvectors for that eigenvalue, which is not possible.
The geometric multiplicity and algebraic multiplicity are both measures of how many times an eigenvalue appears in the characteristic polynomial of a matrix. However, the geometric multiplicity only counts the number of linearly independent eigenvectors, while the algebraic multiplicity counts the total number of times the eigenvalue appears as a root of the characteristic polynomial.
The geometric multiplicity is important because it provides information about the behavior of a matrix and its associated eigenvectors. It can help determine whether a matrix is diagonalizable, and it plays a role in the spectral theorem, which is used to decompose a matrix into its eigenvectors and eigenvalues. Additionally, the geometric multiplicity can be used to understand the long-term behavior of a dynamical system represented by a matrix.