Math 307: Problems for section 3.3

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1. Show that for v, w ∈ Cn
v+w

2

=v

2

+w

2

+ 2 Re( v, w )

and use this to prove the polarization identity
v, w =

1
4

v+w

2

− v−w

2

+ i v − iw

2

− i v + iw

2

2. Show that if q1 , q2 , . . . , qn form a basis in Rn , then they also form a basis when regarded as
vectors in Cn . In other words, show that 1.) if the only linear combination c1 q1 + · · · + cn qn
using real numbers c1 , . . . , cn that equals zero has c1 = · · · = cn = 0, then the same is true for
complex numbers, and 2.) if every vector in Rn can be written as c1 q1 + · · · + cn qn for some
real numbers c1 , . . . , cn then every vector in Cn can be written as a linear combination using
complex numbers. If the basis q1 , q2 , . . . , qn is orthonormal in Rn is is also orthonormal in
Cn ?
3. Show that any 2 × 2 orthogonal matrix is either a rotation matrix or a reflection matrix.
4. Let Q = q1 |q2 | · · · |qk

where q1 , q2 , . . . qk ∈ Rn form an orthonormal set. (That is, they

satisfy qi = 1 for i = 1, . . . , k and qi · qj = 0 if i = j , but there might not be enough vectors
to form a basis, i.e., possibly k < n). Identify the matrices QT Q and QQT . Show that
the projection p of a vector v onto the subspace spanned by q1 , q2 , . . . qk can be written
p = k=1 qi qT v = k=1 (qi · v)qi .
i
i
i
5. For an m×n matrix A with linearly independent columns there is a factorization (called the
QR factorization) A = QR where Q is an m × n matrix whose columns form an orthonormal
set, and R is an upper triangular matrix. For every k = 1, 2, . . . n the first k columns of Q
spans the same subspace as the first k columns of A. In MATLAB/Octave the matrices
Q and R in the QR decomposition of A are computed using [Q R] = qr(A,0). (Without
the second argument 0 a related but different decomposition is computed.)(For those of
you who have learned about Gram-Schmidt: The columns of Q are the vectors obtained
by applying the Gram-Schmidt procedure to the columns of A.
Using MATLAB/Octave, compute and orthonormal basis q1 , q2 for the plane in  4 spanned


R
1
−1
1
1
1
1
by a1 =   and a2 =   Compute the projection p of the vector v =   onto the
1
1
1
1
1
−1
plane. What are the coefficients of p when expanded in the basis q1 , q2 ?

1

6. Using MATLAB/Octave and the discussion in the previous problem, find an orthonormal
  

1
1
0
1 0
0
  

2 1
1
set of vectors q1 , q2 and q3 with the same span as     and  . Provide the commands
0 0
1
  

0 1
1
0
0
0
that you used.
7. Do the following computational experiment. First start with a random symmetric 10 × 10
matrix A (for example B=rand(10,10); A=B’*B; will produce such a matrix) and compute
its QR factorization. Call the factors Q1 and R1 . Now multiply Q1 and R1 in the ”wrong”
order to obtain A2 = R1 Q1 and compute the QR factorization of the resulting matrix A2 .
Repeat this step to obtain a sequence of matrices Qk , Rk and Ak . Do these sequences
converge? If so can you identify the limit? (Hint: eig(C) computes the eigenvalues of C).
8. If U1 and U2 are unitary matrices, is U1 U2 a unitary matrix too?
9. If q1 , . . . , qn is an orthonormal basis for Cn do the complex conjugated vectors q1 , . . . , qn
form an orthonormal basis as well? Give a reason.

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