Concepts

Question 1

What can we say regarding a global/local minimum/maximum for a quadratic form?

Answer 1

If A is positive definite, then x = 0 is the unique global minimum of the quadratic form. This is the case when all leading principle minors are positive

If A is negative definite, then x = 0 is the unique global maximum of the quadratic form. This is the case when the leading principle minors alter in sign.

Question 2

What is positive definite, negative definite, and positive and negative semi definite?

This regards not the special case with quadratic form.

Answer 2

A is positive if every principle minor is non-negative.

A is negative semidefinite if every principal minor of odd order is non-positive and every principal minor of even order is non negative.

Question 3

Vad betyder siffran vi får i vår bordered hessian matrix med constraint?

Answer 3

> 0 så är det en local maximum
< 0 så är det en local minimum

Question 4

I set-teori, vad betyder:

U

∩

Vi har A = 1,2,3
Vi har B = 3,4

Answer 4

U = Union = det är alla punkter som finns i alla set.
A U B = 1,2,3,4

Upp och ned U = ∩ = intersection = De punkter som är lika i båda setten.

A ∩ B = 3

Question 5

What can we say about this set?

A = {x ∈ R : −1 ≤ x ≤ 1}

Answer 5

[−1, 1]

Closed, bounded and compact.

Question 6

What can we say about this set?

B ={x∈R: −1 < x < 1}

Answer 6

(-1,1)

This set is open and bounded

Question 7

C = {x∈R: −1 < x ≤ 1}

Answer 7

(−1,1]

This set is not open or closed, but bounded.

It is not open, because there is no open ε-ball around x = 1 completely contained in C. It is not closed, because its complement (−∞, −1] ∪ (1, ∞) is not open.

Question 8

D = { x in R: x > 0}

Answer 8

D is open.

This set is the interval (0, ∞) on the real number line. It is open, since for every point in the interval, you can move a little in either direction and always remain in the interval. It is not closed, since its complement DC = (−∞, 0] is not open, and therefore it is not compact either.

Question 9

E = {x ∈ R : x ≥ 0}

Answer 9

[0, ∞)

It is closed.

This set is the interval [0, ∞) on the real number line. It is not open, since there is no open ε-ball around x = 0 which is contained entirely within the set. It is closed, since its complement EC = (−∞, 0) is open. However, it is not compact, because it goes off to infinity and there is no upper bound.

Question 10

F ={1,2,3,4,5,…}

Answer 10

Set is closed.

This set consists of all positive integers. It is not open, since you leave the set when moving a small distance away from any integer. It is closed, since its complement FC = 􏰌∞n=1(n,n + 1) ∪ (−∞,1) is an open set. There is no upper bound, so it is not compact.

Question 11

G={1/x :x∈N}∪{0}

Answer 11

This set is closed and compact (since it also is bounded).

Here one should note that it is in N nor R. N means all integers.

This set is given by {1, 1/2, 1/3, 1/4, 1/5, …, 0}.

It is not open; if you move some
small ε from x = 1 for instance, you leave the set. It is closed, since its complement is open. It is also bounded, sinceML = 0 n=1 n n+1
is less than or equal to all numbers in the set, and MU = 1 is greater than or equal to all numbers in the set. Thus, it is compact.

Question 12

G={1/x :x∈N}

Answer 12

This is not open or closed.

This set is given by {1, 1/2, 1/3, 1/4, 1/5, …}.

It is not open, since for example you leave the set when moving a small ε away from x = 1.

It is not closed either. Its complement HC contains zero, and no open ε-ball around zero is entirely contained within HC (since no matter how small you make ε, you can always find some point x1 < ε in the open ε-ball which is an element of H and therefore not an element of HC).

Though it is bounded by the same argument as for G, it cannot be compact since it is not closed.

Question 13

Exercis 8.2 ouch 8.4 handlar också om sets.

Question 14

What is the sandwich formula

Answer 14

AP=PD

Question 15

What is D in terms of P and D?

Answer 15

D = P^{-1}AP

Question 16

Express A in terms of P and D

Answer 16

A = DPP^{-1}

Question 17

Express A^{-1} in terms of P and D

Answer 17

PD^{-1}P^{-1}

Question 18

Move A to LHS

ABC = DE

Move E to LHS

ABC = DE

Premultiply A^{-1}

ABC = DE

Answer 18

BC = A^{-1}DE

ABCE^{-1}=D

A^{-1}ABC=A^{-1}DE

or

ABCA^{-1}=DEA^{-1}

Question 19

Trace(A) =

A = any matrix

Answer 19

Trace(A) = Trace(D)

D = The diagonal vector of eigen values

Question 20

Det(D)

Determinant of a matrix A

Answer 20

Det(D) = Det(A)

Question 21

What can you do with trace()

Answer 21

One can set Trace(A) = Trace(D) and solve for a variable a that is in A.

Question 22

Vad är trace() ?

Answer 22

Summan av de diagonala numrern i en matris.

Question 23

Det finns Hessian, borderd hessian och borderd matrix…. vilken är vilken och vad kan man säga.

Answer 23

KOLLA DETTA PÅ YOUTUBE. Verkar finnas bra filmer.

If the bordered hessian matrix is > 0, it is a local maximum.

If the bordered hessian matrix is < 0, it is a local minimum.

Dubbelkolla detta

If the hessian matrix is > 0, it is a local minimum.

If the bordered hessian matrix is < 0, it is a local minimum.

Question 24

{x ∈ R: x > 0}

Answer 24

Open, not bounded.

Question 25

{x ∈ R: −∞ < x <∞}

Answer 25

Both open and closed, not bounded.

Question 26

{x ∈ R: −∞ < x <∞}

Answer 26

Both open and closed, not bounded.

Question 27

{(x,y) ∈ R^2: xy=1}

Answer 27

Closed, not bounded.

Question 28

{(x,y) ∈ R^2: xy≠0}

Answer 28

Open! This, since it includes all the points in R^2 except the x and y axis.

Question 29

{(x,y) ∈ R^2: xy≠0}

Answer 29

Open! This, since it includes all the points in R^2 except the x and y axis.

Question 30

{(x,y) ∈ R^2: 4 ≤ x^2 + y^2 ≤ 9}

Answer 30

Closed!

Question 31

{(x,y) ∈ R^2: 0 < x ≤ 1 and 0 < y ≤ 1}

Answer 31

Neither!

Question 32

{(x,y) ∈ R^2: −∞ < x <∞ and −∞ < x <∞}

Answer 32

Both open and closed

Question 33

What is the intersection and union of A = {x ∈ R : −1 ≤ x ≤ 1} and D = { x in R: x > 0}. Open, closed?

Answer 33

A ∩ D = (0, 1]. This set is neither open nor closed. A ∪ D = [−1, ∞). This set is closed. It is not open.

Question 34

What is the intersection and union of A = {x ∈ R : −1 ≤ x ≤ 1} and E = {x ∈ R : x ≥ 0} Open, closed?

Answer 34

A ∩ E = [0,1]. This set is closed. It is not open. A ∪ E = [−1, ∞). This set is closed. It is not open.

Question 35

What is the intersection and union of B ={x∈R: −1 < x < 1} and D = { x in R: x > 0}. Open, closed?

Answer 35

B ∩ D = (0,1).This set is open. It is not closed. B ∪ D = (−1, ∞). This set is open. It is not closed.

Question 36

What is the intersection and union of B ={x∈R: −1 < x < 1} and E = {x ∈ R : x ≥ 0}

Answer 36

B ∩ E = [0, 1). This set is neither open nor closed. B ∪ E = (−1, ∞). This set is open. It is not closed.

Question 37

What os supremum and infimum?

Answer 37

A set is bounded if it is bounded both from above and below. The supremum of a set is its least upper bound and the infimum is its greatest upper bound.

Question 38

When is a multivariate function convex?

Answer 38

If and only if the hessian with second order derivatives is positive semidefinite. The hessian is positive semidefinite if and only if all principal minors are non-negative.

Question 39

What can we say regarding (strict) concavity and (strict) convexity regarding the hessian matrix?

Answer 39

Note this is different then with the quadratic form regarding signs! When all leading principal minors are > 0, the hessian matrix is positive definite and the function is strictly convex. When all leading principal minors are < 0, the hessian matrix is negative definite and the function is strictly concave. This implies that the function also is, concave, strict quasi concave and quasi concave. When the all the principal minors are ≥ 0, the hessian matrix is positive semidefinite and the function is convex. When the all the principal minors are ≤ 0, the hessian matrix is negative semidefinite and the function is concave. This implies that the function is quasi concave.

Question 40

How should one best evaluate if a function is (strict) concave and (strict) convex?

Answer 40

The best way is to start looking at the leading principal minors (LPM) of the function. If we have positive (negative) definiteness, then the function is strict convex (concave). (Note this is the opposite as with the the quadratic form regarding signs). Because than we know for sure that the function is convex (concave). If we have alternating signs, we most look at all the principal minors. Here, convexity (concavity) do not imply strict concavity (convexity), but the other way goes.

Question 41

If f(x) is concave, then -f(x) is....

Answer 41

convex.

Question 42

If f(x) is strictly concave, then we know f(x) is...

Answer 42

concave.

Question 43

If the function is (strict) quasi concave....we can

Answer 43

not conclude that the function is concave.

Question 44

if the function is concave...we can

Answer 44

conclude that the function is quasi concave.

Question 45

How do we find out that a function is (strict) quasi concave? Do this always hold?

Answer 45

We look at the full bordered hessian matrix (bordered with the first order derivatives). If |H^| ≥ 0, the function is quasi concave. If If |H^| > 0 it is strictly quasi concave. This do only hold for the two variables case. With more than two variable, this is only sufficient conditions. Thus, if |H^| ≥ 0, we can not say for sure that the function is quasi concave if we have three or more variables. But, we can reject quasi concavity in the same setting if |H^| < 0. But, if |H^| > 0 (i,e, strictly larger than o), then the function is strictly quasi concave.

Question 46

Show the expressions for a concave and for a convex function, using inequalities

Answer 46

Concave: f(λx + (1-λ)y) ≥ λf(x) + (1-λ)f(y) Convex: f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y)

Question 47

Vad är derivatan av ln(x)

Answer 47

1/x

Question 48

Vad är derivatan av a^x

Answer 48

a^x ln a

Question 49

Vad är derivatan av e^kx

Answer 49

ke^kx

Question 50

Vad derivatan av e^{x+y}

Answer 50

e{x+y}

Question 51

vad är derivatan av ln(x^2+y)

Answer 51

2x/(x^2+y)

Question 52

vad är derivatan av e^x^2

Answer 52

2xe^x2

Question 53

Vad är derivatan av ln(x+y+5)

Answer 53

1/(x+y+5)

Question 54

Write the expression for taylor polynomial

Answer 54

See notes

Question 55

Write the expression for taylor polynomial in matrix form

Answer 55

See notes

Question 56

What will a quasi concave function guarantee?

Answer 56

That we will have a local maximum.

Question 57

What does the implicit function theorem say? f(x,z,y)

Answer 57

If z can be implicitly defined by x, then when get the change in z due to x by -dx/dz, which is equal to -df/dz/df/dx

Question 58

Write an expression for A^m using APPD

Answer 58

A^m = PD^mP^-1