Let P =Thena)P has two linearly independent eigenvectorsb)P has an eigen vectorc)P is non singulard)There exists a non singular matrix S such that S-1PS is a diagonal matrix.Correct answer is option 'B'. Can you explain this answer?

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Let P =

Then

a)
P has two linearly independent eigenvectors
b)
P has an eigen vector
c)
P is non singular
d)
There exists a non singular matrix S such that S^-1PS is a diagonal matrix.

Correct answer is option 'B'. Can you explain this answer?

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Let P =Thena)P has two linearly independent eigenvectorsb)P has an eig...

eigenvalues are given by |P — λI| = 0
λ = 0,0 not distinct
=>P is not diagonalizable

=> singular (also, |P| = 0, product of eigenvalues)
eigenvector corresponding to λ = 0 is given by

X₁ + ix₂ = 0 after apply R₂ → R₂ — iR₁
P has only one eigenvector
P is diagonalizable if algebraic multiplicity and geometric multiplicity of eigenvalues are equal. λ = 0 has algebraic multiplicity 2 whereas its geometric multiplicity is 1.
Hence, P is not diagonalisable.

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Let P =Thena)P has two linearly independent eigenvectorsb)P has an eigen vectorc)P is non singulard)There exists a non singular matrix S such that S-1PS is a diagonal matrix.Correct answer is option 'B'. Can you explain this answer?

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