Let T be a linear operator on a finite dimensional space V and C is any scalar then C is characteristic value of T ifa)the operator (T - Cl) is singularb)the operator (T - CI) is non-singularc)the operator (T - CI) is identitd)he operator (T - CI) is zero.Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Explanation:

To understand why option 'A' is the correct answer, let's go through each option and analyze its implications.

a) The operator (T - Cl) is singular:
If the operator (T - Cl) is singular, it means that the linear transformation (T - Cl) does not have an inverse. In other words, there exists a non-zero vector v such that (T - Cl)v = 0.

If (T - Cl)v = 0, then Tv - Clv = 0, which can be rewritten as Tv = Clv. Notice that this equation is in the form of a characteristic equation, where C is the characteristic value and v is the corresponding eigenvector. Therefore, if (T - Cl) is singular, C is a characteristic value of T.

b) The operator (T - CI) is non-singular:
If the operator (T - CI) is non-singular, it means that the linear transformation (T - CI) has an inverse. In other words, for any non-zero vector v, there exists a unique vector u such that (T - CI)u = v.

If (T - CI)u = v, then Tu - Cu = v, which can be rewritten as Tu = Cu + v. Notice that this equation does not have the same form as a characteristic equation, since the right-hand side does not involve the scalar C alone. Therefore, having a non-singular operator (T - CI) does not necessarily imply that C is a characteristic value of T.

c) The operator (T - CI) is identit:
If the operator (T - CI) is the identity operator, it means that for any vector v, (T - CI)v = v. This implies that Tv - Cv = v, which can be rewritten as Tv = Cv + v. Again, this equation does not have the same form as a characteristic equation, since the right-hand side does not involve the scalar C alone. Therefore, having the identity operator (T - CI) does not necessarily imply that C is a characteristic value of T.

d) The operator (T - CI) is zero:
If the operator (T - CI) is zero, it means that for any vector v, (T - CI)v = 0. This implies that Tv - Cv = 0, which can be rewritten as Tv = Cv. Notice that this equation is in the form of a characteristic equation, where C is the characteristic value and v is the corresponding eigenvector. Therefore, if (T - CI) is zero, C is a characteristic value of T.

Based on the explanations above, we can conclude that option 'A' is the correct answer, as having the operator (T - Cl) singular implies that C is a characteristic value of T.

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