Let u = (1, 2, 2, 4). Use an elementary projector to find an orthonormal basis in R4 that contains the unit vector u.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1.   Let u = (1, 2, 2, 4). Use an elementary projector to find an orthonormal basis in R4 that contains the unit vector u.

Suppose S Rn×n  and TRn×n are two elementary reflectors that are not identical. Under which condition do we have ST = TS?

Hint:  Wrire S = I2aaT and  T = I = 2bbT for some unit vectors a, bRn. Then find conditions for a, b such that ST = TS.

3. Let m, n be positive integers. Let R ∈ Rn×n be an elementary reflector. Is S = Im×m 0 (m+n)×(m+n) an elementary reflector?

4.Let A ∈ Rm×n  with rank r = rank(A). Suppose P ∈ Rm×m  is an orthogonal matrix such that PT A =  where A1 ∈ Rr×r is upper triangular with nonzero diagonal entries and A2 ∈ Rr×n.

Show that the first r columns of P  is an orthonormal basis for R(A) and the last columns of P is an orthonormal basis for N (AT ).

Hint: left-multiply both sides by P , then express columns of A as linear combinations of columns of P , and use the orthogonal complement R(A)⊥ = N (AT ).

Use Problem 4 and Householder reduction (see the lecture note Orthogonal Reduction) to obtain orthonormal bases for R(A) and N (AT ). You may use software for matrix computations but the function qr() in Matlab is not allowed.

Practice Problems

(For your practice only! Do not submit these problems as they will not be graded)

1. Let n Use the Gram-Schmidt procedure to obtain a QR factorization for L (You may use software for the computations).

2. Use Householder reduction to find a QR factorization of L (You may use software for the computations).

3. Use Givens reduction to find a QR factorization of L (You may use software for the compu- tations).

4. Let Similar to problem 5 above, obtain orthonormal bases for R(AT ) and N (A). Then use the results to construct a URV decomposition for A. (You may use software for the computations).

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