Sample Problems
1.
For any
matrix A, the trace of A is defined by
(a)
Prove that
for all
matrices A and B.
(b)
Prove that, for all
matrices A and B and all scalars
and
,
(c)
Prove that an
matrix A satisfies
if and only if A=0.
2.
Let A be an
matrix. Prove that
T
3.
T
Here Null(A) denotes the null space (kernel) of A, while Col(A ) is the column space (range) of A .
The
(a)
(b)
matrix A is given by
Find the Singular Value Decomposition (SVD) of A.
Find
, the Moore-Penrose generalized inverse of A(you can express it in factored form, if
(c)
4.
convenient).
Find the least-squares solution of Ax=b, where
Let V be the vector space of real polynomials of degree less than or equal to 2. Define an inner
product
on V by
(a)
Use the Gram-Schmidt (or modified Gram-Schmidt) procedure to produce an orthogonal basis
for V from the standard basis
.
(b)
2
5.
Find the coordinates of p(x)=6x in the orthogonal basis you just computed.
The set
, where
is an orthogonal basis for
, while
is an orthogonal basis for
. The
, where
matrix A is defined by
Find orthogonal bases for:
(a)
Null(A)
(b)
T
(c)
(d)
Null(A )
Col(A)
T
Col(A )
6.
Suppose A is an
(a)
(b)
7.
Prove that the eigenvalues of A are real, and the corresponding eigenvectors can be chosen to
be real.
Prove that eigenvectors of A corresponding to distinct eigenvalues are orthogonal.
Let A be an
8.
real symmetric matrix.
real symmetric matrix. Prove that there is an orthonormal basis for
consisting of eigenvectors of A. (You may use the results of the previous exercise.)
T
Let A be a real
T
matrix. Prove that AA and A Ahave the same nonzero eigenvalues.
9.
2
Let P be a symmetric
matrix satisfying P =P, and assume that P is neither the zero matrix
nor the identity matrix. Let W be the column space of P and W be the null space of P.
1
2
(a)
Prove that
if and only if P x=x.
(b)
(c)
Prove that if
is an eigenvalue of P, then
Prove that
is the direct sum of W and W . That is, prove that every
1
written uniquely as x=y+z,
,
is zero or one.
2
can be
.
(d)
Prove that, for each
, P x is the vector in W closest to x (in the Euclidean norm).
1
10.
Let A be an
matrix. Prove that eigenvectors corresponding to distinct eigenvalues are linearly
independent. That is, prove that if
are corresponding eigenvectors, then
are distinct eigenvalues of A, and
is a linearly independent set.
11.
Suppose
and
are two different orthonormal bases for a subspace W of
. Suppose further that the scalars a , i,j=1,2,3, satisfy
ij
(a)
Show that the
matrix A whose entries are a , i,j=1,2,3, is orthogonal.
ij
(b)
T
T
T
+u u +u u
1 1
2 2
3 3
Prove that the matrices u u
T
T
T
+v v +v v
1 1
2 2
3 3
and v v
are equal.
(c)
T
T
T
+u u +u u :
1 1
2 2
3 3
Interpret the action of P=u u
12.
If
, what is the significance of Px?
Let V be an inner product space, let W be a finite-dimensional subspace of V with basis
, and let v be any vector in V.
(a)
Prove that there is a unique vector
(b)
13.
closest to v (the best approximation to v from W).
``Closest'' is defined in terms of the norm induced by the inner product.
Derive the normal equations for computing the best approximation w to v from W.
Give an example to show that Gaussian elimination without partial pivoting can be unstable in finite
precision arithmetic. Show that the use of partial pivoting eliminates the instability in your example.
(Hint: The matrix need not be large--a
matrix will do!)
14.
Suppose A is a nonsingular
Give a bound on
matrix and
in terms of
satisfy
.