8: Suppose that A and B are similar matrices, B = p-1AP, where 6 2 P...
Let А and B be similar nxn matrices. That is, we can write A = CBC- for some invertible matrix с Then the matrices A and B have the same eigenvalues for the following reason(s). A. Both А and A. Both А and B have the same characteristic polynomial. B. Since A = CBC-1 , this implies A = CC-B = IB = B and the matrices are equal. C. Suppose that 2 is an eigenvalue for the matrix B...
Two n x n matrices A and B are called similar if there is an invertible matrix P such that B = P-AP. Show that two similar matrices enjoy the following properties. (a) They have the same determinant. (b) They have the same eigenvalues: specifically, show that if v is an eigenvector of A with eigenvalue 1, then P-lv is an eigenvector of B with eigenvalue l. (c) For any polynomial p(x), P(A) = 0 is equivalent to p(B) =...
Are the two matrices similar? If so, find a matrix P such that B = p-1AP. (If not possible, enter IMPOSSIBLE.) 200 100 B = A= 0 3 0 03 0 0 0 1 0 0 2 0 0 ра 0 11 X
Linear Algebra Suppose that B=P-1AP. (a) Prove that A and B have the same eigenvalues. (b) Prove that if x is an eigenvector of A, then P-10 is an edigenvector of B.
Are the two matrices similar? If so, find a matrix P such that B = p-1AP. (If not possible, enter IMPOSSIBLE.) [200 1 0 0 A= 030 B= 0 30 0 0 1 0 0 2 P= 11
Are the two matrices similar? If so, find a matrix P such that B = P-1AP. (If not possible, enter IMPOSSIBLE.) 3 00 1 0 0 A = 0 2 0 0 30 0 0 1 0 0 2 P=
Are the two matrices similar? If so, find a matrix P such that B = p-1AP. (If not possible, enter IMPOSSIBLE.) [200 [100 0 3 0 B = 0 3 0 A = 0 0 1 0 0 2 P= III
let a and b be n*n similar matrices, namely, B=S^-1 AS. show that the matrices a and b have the same characteristic polynomial, det(a-λI)=det(b-λI) and, consequently, the same eigenvalues.
3. Let A and B be any nxn matrices. Suppose ū is an eigenvector of A and A+B with corresponding eigenvalues 1 and p. Show that ū is also an eigenvector for B and find an expression for its corresponding eigenvalue. [2]
oru 2 Let A and B be two n x n matrices. There exists a nonsingular matrix P such that PB = AP. Then which of the following is always true? a) A and B are not similar b) A and B have the same eigenvalues c) A does not have any characteristic polynomial d) B does not have any characteristic polynomial