Statistics Solutions

[See Steps] (a) Two n x n matrices A and B are said to be similar if there exists a nonsingular matrix C such that B=C^-1 A C Prove that if A and B are

Question: (a) Two n x n matrices \(\mathrm{A}\) and \(\mathrm{B}\) are said to be similar if there exists a nonsingular matrix C such that

\[B=C^{-1} A C\]

Prove that if \(\mathrm{A}\) and \(\mathrm{B}\) are similar matrices, they have the same eigenvalues.

Hint: If \(\lambda\) is an eigenvalue of \(B\) then, by definition,

\[\left|B-\lambda I_{n}\right|=0\]

(b) Prove that if A is an idempotent matrix, the eigenvalues of A are either 0 or 1 .

(c) Let

\[M=I_{n}-X\left(X^{\prime} X\right)^{-1} X^{\prime}\]

where \(X\) is an n x k matrix of rank \(\mathrm{k}\). Prove that

\[\operatorname{tr}(M)=n-k\]

Hint: Use the properties of the trace operator stated in lectures.

Price: $2.99
Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document


∴ Other downloads you may be interested in ∴

log in to your account

Forgot password?
Don't have a membership account?
REGISTER

reset password

Back to
log in

sign up

Back to
log in