Consider a matrix A with the special property that Am TI for some positive integ

Question

Consider a matrix A with the special property that Am TI for some positive integer m. Such a matrix is special, as such we might expect it to have special eigenvalues Say v is an eigenvector of A with eigenvalue t. By left-multiplying m times by the matrix A we get Since v is an eigenvector with eigenvalue t, this implies that Since eigenvectors are non-zero the only way this can be satisfied is if trn v = v. 7m implying that the eigenvalues of A must be mth roots of unity An example of such a matrix is 0 -1 Here m = | sqrt(6) | (please enter the smallest possible m) and the eigenvalues are the set .11 Recall: . a set in Maple notation is something of the form (1,1+2*I,3-2*I) . remember to use capital i (1) to represent V2 can be entered using the Maple syntax sgrt (2)

Explanation / Answer

The given matrix B is special matrix with property B^m =I. The given eigen values 1 and -1.

To choose the smallest m value we can use t^m=1 where t is the eigen value

So by using eigen value given 1^m=1 , the smallest m value satisfies this condition is 1.

If the eigen value is -1 then (-1)^m =1 here the m value satisfies the condition is 2

So from the above two values the smallest m value is 1.

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.