is this correct?
B = CAC^(-1)
B-λIn = CAC^(-1) - λIn
using the hint: λIn = CλInC^(-1)
B-λIn = CAC(^-1) - CλInC^(-1)
B-λIn = C[ AC^(-1) - λInC^(-1) ] factored out C
B-λIn = CC^(-1)( A-λIn)
B-λIn = A-λIn, CC^(-1) cancel each other out
therefore det(B-λIn) = det(A-λIn)...
well i know det(ƛIn - A)= 0
thus giving the characteristic polynomial (ƛ-ƛ1)(ƛ-ƛ2)...(ƛ-ƛn)
do i set set det(ƛIn - A) = det(ƛIn - B)?
with the given vector v: Av = ƛv and Bv = ƛv
therefore Av = Bv?
Homework Statement
Suppose that C is an invertible matrix, and you are told that
B = CA(C^-1)
prove that A and B have exactly the same characteristic polynomial
do not assume A and B are triangular or diagonal matrices
Homework Equations
given hint: explain why ƛIn =...
we are given B = CAC^-1
Prove that A and B have the same characteristic polynomial
given a hint: explain why ƛIn = CƛInC^-1
what I did was:
B = CAC^-1
BC = CA
Det(BC) = Det(CA)
Det(B) Det(C) = Det(C) Det(A)
Now they’re just numbers so I divide both sides by Det(C)
Det(B) = Det(A)...