Hi could you help me solve problem 4,5, 6 in the photo I attached? this is a linear algebra course. Please write the answers as detailed as possible. Thanks!
Problem 4: Angle between vectors
17 points. We consider two unit vectors u and v. Calculate the eigenvalues of the matrix
uu- vu
in terms of the angle between the two vectors.
Problem 5: Rank-one update of QR decomposition
15 points. Suppose a matrix A € Rnxn has full rank and we have already obtained its QR decomposition
A = QR. Now, the matrix B = A + uut is the rank-one update of A, where u, v € R” are two vectors.
Prove that, if u’A-‘u #-1, then B is invertible. Design an algorithm that solves the equation Bx = b in
O(n) operations.
Problem 6: Gershgorin disc
10 points. Assume that
Αχ = λx
Show that there exist i and j such that
12-ais laik! | 1 laul
x 0
k.ku
1,1#]
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