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Is it true that if eigen value exists then we will always have a non-zero linearly independent vector ?

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For a 3*3 identity matrix , we have eigen value as 1 but there is no linearly independent vector for it since if they are 3 unknowns then we shall have x=y=z=0 so then how is it true that whenever we have an eigen value existing then we have a non-zero linearly independent vector for it ?

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