Question

Let T: R³ → R³ be an R-homomorphism. Given (1,-1,2) are the eigen values of T, such that E₁(T)= {span}(5,2,1) E₋₁(T)= {span}(0,0,7). Let T²+I (where I is the identity map). Then L:R³ → R³ is an F-Homomorphism.

Find all eigenvalues of L.

Answer 1

To find the eigenvalues of linear map L which is defined as T² + I, square the eigenvalues of T and add 1. The eigenvalues of L are 2 with multiplicity 2, and 5.

The student is tasked with finding the eigenvalues of a linear map L which is given in terms of another linear map T. Since T has eigenvalues 1, -1, and 2 and L is defined as T² + I, where I is the identity map, the eigenvalues of L can be found by evaluating each eigenvalue of T, squaring it and then adding 1 (because of the identity map addition). Therefore, if λ is an eigenvalue of T, then λ² + 1 will be an eigenvalue of L.

Using this process, for T having eigenvalues 1, -1, and 2, the corresponding eigenvalues of L will be:

• (1)² + 1 = 2
• (-1)² + 1 = 2
• (2)² + 1 = 5

So, the eigenvalues of L are 2 (with multiplicity 2) and 5.