If A, B,C are 3 n times n square matrices show that Tr(ABC) = Tr(CAB) = Tr(BCA),
ID: 2060834 • Letter: I
Question
If A, B,C are 3 n times n square matrices show that Tr(ABC) = Tr(CAB) = Tr(BCA), where Tr denotes the trace of a matrix. i. e. the sum of its diagonal elements. Show that the trace of a matrix remain the same (i.e invariant) under a unitary transformation. Let A be an n times n square matrix with eigenvalues v1, v2, ..., vn. Show that |A| = v1v2 ... vn and hence that the determinant of A is another invariant property. Show that if A is Hermitian, then U = (A + iI)(A - iI)-1 is unitary. (I here is the identity matrix.) Show that |I + A| = I + TrA + O where A is an n times n square matrix. Show that where A is a n times n square matrix.Explanation / Answer
too many question, i only can help you solve one
I+A = (A + I)^2 = I + I + A +2A = A(T) + I