Question

Let A be a symmetric matrix of size 3 × 3 and consider the following quadratic function f(x) = x T Ax. Show that the gradient of f is: ∇xf(x) = 2Ax. A matrix is symmetric if Aij = Aji for all i and j.

Answer #1

Let Q(t) be a matrix, function of time, t. Assume that Q is
orthogonal for all t. Show that if Q(t_0) = I (identity matrix) for
some t_0 then d/dt (Q(t_0)) is skew-symmetric.

Let A be a given (3 × 3) matrix, and consider the equation Ax =
c, with c = [1 0 − 1 ]T . Suppose that the two vectors
x1 =[ 1 2 3]T and x2 =[ 3 2 1] T are
solutions to the above equation.
(a) Find a vector v in N (A).
(b) Using the result in part (a), find another solution to the
equation Ax = c.
(c) With the given information, what are the...

(a)
For the following quadratic forms gi(x) write down the
associated symmetric matrix Ai such that gi(x) = xT Aix.
g1(x1, x2) = x21 − 2x1x2 + 4x2
g2(x1, x2) = 4x21 − 6x1x2 + x2
g3(x1, x2, x3) = 3x21 + 3x2 + 5x23 + 2x1x2 − 2x1x3 − 2x2x3
g4(x1, x2, x3) = −3x21 − x2 + 8x2x3 − 16x23
(b) Determine the definiteness of g1(x) and g3(x) using the
method of eigenvalues.
(c) Determine the definiteness of...

Problem 3.2
Let H ∈ Rn×n be symmetric and idempotent, hence a projection
matrix. Let x ∼ N(0,In). (a) Let σ > 0 be a positive number.
Find the distribution of σx. (b) Let u = Hx and v = (I −H)x and ﬁnd
the joint distribution of (u,v). 1 (c) Someone claims that u and v
are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ.
(e) Assume that 1 ∈ Im(H) and ﬁnd...

Let B = [ aij ] 20×17 be a matrix with real entries. Let x be in
R 17 , c be in R 20, and 0 be the vector with all zero entries.
Show that each of the following statements implies the other.
(a) Bx = 0 has only the trivial solution x = 0 n R 17, then (b)
If Bx = c has a solution for some vector c in R 20, then the
solution is unique.

a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1
≤ i, j ≤ n) and having the following interesting property:
ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n
Based on this information, prove that rank(A) < n.
b) Let A ∈ R m×n be a matrix of rank r. Suppose there are right
hand sides b for which Ax = b has no solution,...

Consider W as the set of all skew-symmetric matrices of size
3×3. Is it a vector space? If yes, then ﬁnd its dimension and a
basis.

Consider the following.
f(x, y) = x/y, P(4,
1), u =
3
5
i +
4
5
j
(a) Find the gradient of f.
(b) Evaluate the gradient at the point P.
(c) Find the rate of change of f at P in the
direction of the vector u.

n x n matrix A, where n >= 3. Select 3 statements from the
invertible matrix theorem below and show that all 3 statements are
true or false. Make sure to clearly explain and justify your
work.
A=
-1 , 7, 9
7 , 7, 10
-3, -6, -4
The equation A has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation x → Ax is one-to-one.
7. The equation Ax...

Let X be the set {1, 2, 3}.
a)For each function f in the set of functions from X to X,
consider the relation that is the symmetric closure of the function
f'. Let us call the set of these symmetric closures Y. List at
least two elements of Y.
b) Suppose R is some partial order on X. What is the smallest
possible cardinality R could have? What is the largest?

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