Question
eA denotes the exponential of a square matrix A. Suppose λ is an eigenvalue and v is the corresponding eigen-vector of matrix A.
Consider the following two statements :
Statement 1 : eλ is an eigenvalue of eA .
Statement 2 : v is an eigen-vector of eA .
Which one of the following options is correct?
Options :
Statement 1 is true and statement 2 is false
Statement 1 is false and statement 2 is true.
Both the statements are correct
Both the statements are false.
Answer :
Both the statements are correct
Solution :
Given : Exponential series,
eA= 1+ A/1! + A2/2! + A3/3! +.......∞
eA= 1+ A/1! + A2/2 + A3/6 +.......∞
Polynomial function of infinite series
where, An×n= Matrix, λA = Eigen value of A and V= Eigen vector corresponding to λA
Any operation on matrix, same operation are performed on eigen values, so eλ will the eigen value of eA but eigen vector remains same.
So, V will be an eigen vector of eA
That means, both statements are correct.
Hence, the correct option is (C).
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