Reference no: EM133407161

Question 1. Let A ∈ C^n×n be an invertible matrix.

(a) Prove that A^∗ is invertible and (A^∗)-1 = (A^-1)^∗.
(b) Define B = A^-1A∗. Prove that B is unitary if and only if A is normal.

Question 2. Consider the linear transformation T : P₃(R) → R^4×1 defined by

 

For example,

 

Throughout this problem, use the standard inner product, addition, and scalar multiplication for all vector spaces involved.
(a) Find an orthogonal basis B for R(T).
(b) Find the best approximation for (1, 1, -1, 1)^T in R(T).
(c) Find a least squares solution of T (x ) = (1, 1, -1, 1)^T.
(d) Find an orthogonal basis C for R^4×1 such that B is a subset of C.

Question 3. Let A ∈ C^3×3 such that A³ is the zero matrix but A² is not the zero matrix. Find the Jordan matrix similar to A.

Question 4. Let w ∈ R^n×1 with ?w?₂ = 3. Define A = 1 - ww^T where I is the n × n identity matrix.

(a) Show that w is an eigenvector of A and find its corresponding eigenvalue.
(b) Let v R^n×1 such that v is orthogonal to w. Show that v is an eigenvector of A and find its corresponding eigenvalue.
(c) Find Σ such that QΣP* is a singular value decomposition of A. You must specify the exact entries of Σ to receive full credit.
(d) Prove that A is diagonalizable.
(e) Is A invertible? Make sure to justify your answer.