Math 307: Problems for section 3.3
KnowledgeCats1. 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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