The geometric multiplicity of an eigenvalue is always greater than or equal to 1 because an eigenvector is defined as a nonzero vector that satisfies the equation (A - λI)x = 0. This definition ensures that the eigenspace, which is the span of all eigenvectors, cannot be limited to just the null vector. Therefore, the presence of at least one nonzero eigenvector guarantees that the geometric multiplicity is at least 1.
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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?