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The geometric multiplicity of an eigenvalue is the dimension of the linear space of its associated eigenvectors (i.e., its eigenspace). In this lecture we provide rigorous definitions of the two concepts of algebraic and geometric multiplicity and we prove some useful facts about them.
Algebraic and Geometric Multiplicity. Compute the characteristic polynomial, det(A its roots. These are the eigenvalues. For each eigenvalue. , compute Ker(A. -eigenspace, the vectors in the. tId), and. nd. Id).
Algebraic multiplicity vs geometric multiplicity. The geometric multiplicity of an eigenvalue λ of A is the dimension of E A ( λ). In the example above, the geometric multiplicity of − 1 is 1 as the eigenspace is spanned by one nonzero vector.
Let's suppose there are 3 eigenvalues: , , . Let the eigenspaces be U , V. and W . Let the geometric multiplicities be a = dim U , b = dim V , c = dim W , with a + b + c = n. Let ~ u1, ~u2, . . . , ~ ua. be a basis of U ; let ~ v1, ~v2, . . . , ~vb. be a basis of and let ~w1, ~w2, . . . , ~wc.
May 24, 2024 · T he geometric multiplicity of an eigenvalue of algebraic multiplicity n is equal to the number of corresponding linearly independent eigenvectors. The geometric multiplicity is always less than or equal to the algebraic multiplicity.
The algebraic multiplicity of λi to be the degree of the root λi in the polynomial det (A − λI). The geometric multiplicity the be the dimension of the eigenspace associated with the eigenvalue λi. For example: [1 1 0 1] has root 1 with algebraic multiplicity 2, but the geometric multiplicity 1.
the algebraic multiplicity of = 3 is 2, but the geometric multiplicity of = 3 is 1. Both = 3 and = 5 have algebraic multiplicity 1 and geometric multiplicity 1. Example: Let A= 3 2 1 4 , as in our previous examples. Then both = 2 and = 5 have algebraic multiplicity 1 and geometric multiplicity 1. Example: Let C= 2 4 2 0 0 4 2 0 6 0 2 3 5. Then ...
