Suppose a is a real symmetric matrix if the eigen-values of

Question

Part A -

(a) Suppose A is a real symmetric matrix. If the eigen-values of A equal to only 0 and 1 then prove that A is idempotent.

(b) Let A be a real symmetric idempotent matrix. Show that tr(A = rank(A).

(c) Let A be a square matrix, and U is an orthogonal matrix. Prove that tr(U^TAU) = tr(A).

Part B -

Consider random variables X₁, . . . , X_K. They are not necessarily independent. E(X_j) = μ_j and Var(X_j) ≤ σ² < ∞ or all j = 1, . . . , K. We are interested in bounding from below the probability P(max_1≤j≤K{|X_j - μ_j|} ≤ t).

(a) Use the property

P(max_1≤j≤k{|X_j - μ_j|} ≤ t) = 1 - P(∃_j : |X_j - μ_j| > t)

the union bound

P(∃_j : |X_j - μ_j| > t) ≤ _j=1∑^K P(|X_j - u_j| > t),

And the Chebyshev's inequality for each summand:

P(|X_j - μ_j| > t) ≤ E(|X_j - μ_j|²)/t² ≤ σ²/t²

To derive a lower bound on P(max_1≤j≤K{|X_j - μ_j|} ≤ t) depending on t, K and σ².

(b) Write a lower bound on P(max_1≤j≤K{X_j} ≤ t) under the additional condition: E{exp(X_jh)} < ∞ for all h > , using the exponential Chebyshev's inequalities for each X_j.