Recently, I got to know that, to be diagonalizable, the geometric multiplicity should be same with algebraic multiplicity.
Is there any general condition for geometric multiplicity to be lower than algebraic multiplicity?
2026-04-04 11:49:05.1775303345
General condition for geometric multiplicity to be lower than algebraic multiplicity.
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An eignvalue $\mu$ always has geometric multiplicity lower or equal than the algebraic multiplicity. If the power in which $(x-\mu)$ appears in the minimal polynomial of the transformation is greater than 1 then the geometric multiplicty is stricly lower.
If you don't know, the minimal polynomial of an endomorphism\matrix A is the polynomial $m$ of least degree such that $m(A)$=0. All eignvalues are roots of this polynomial, as well as of the characteristic polynomial.