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EXERCISES 21-6-2014

Ex. 0.1. Let v =

 1 2

−1

 and w =

 1 1 1

 Find (if any) all matrices A ∈ M

3,3

(R) such that: T r(M ) = 4, M v = 2v, M w = −w and A has an eigenvector orthogonal both to v and w.

Ex. 0.2. (a) Let v = 1 2



. Find (if any) all symmetric matrices A ∈ M

2,2

(R) such that −3 is an eigenvalue of A and Av = 2v.

(b) Let w =  1

−1



. Is there a symmetric matrix B ∈ M

2,2

(R) such that Av = 2v and Aw = 4w?

(c) Is there a hermitian matrix E ∈ M

2,2

(C) such that Av = 2v and Aw = 4w?

Ex. 0.3. (a) Let C be an orthogonal matrix. Prove that det C = +1 or det C = −1. (Hint: use the fact that, for any square matrix, det A = det A

t

. (b) Let B = {A

1

, A

2

, A

3

, A

4

} be an orthogonal basis of V

4

such that k A

1

k=

3, k A

2

k= 5, k A

3

k= 2, k A

4

k= 2. Let A = (A

1

A

2

A

3

A

4

) be the matrix whose columns are A

1

, A

2

, A

3

, A

4

. Compute | det A|.

Ex. 0.4. Let B = {(1, 2), (1, 3)}. Sia T : V

2

→ V

2

be the linear transfor- mation such that m

BB

(T ) =  9 13

−6 −9



. Moreover, let S : V

2

→ V

2

be the linear transformation such that m

BB

(S) = 1 3

3 4



. Is T a symmetric linear transformation?. Is S a symmetric linear transformation?.

Ex. 0.5. Let v = (1, 2, 1). Let us consider the function T : V

3

→ V

3

defined as follows: T (X) = X × v.

(a) Prove that T is a linear transformation . (b) Is T symmetric?

(b) Compute the matrix representing T with respect to the canonical basis of V

3

and compare the result with the answer given to (a)

(c) Find an orthonormal basis of V

3

B = {v

1

, v

2

, v

3

} with v

1

parallel to v and compute m

BB

(T ).

Ex. 0.6. Let A =

1 0 0 0

0 −1 0 0

0 0 0 1

0 0 1 0

. Let us consider the real quadratic form

Q(X) = XAX

T

.

1

(2)

2 EXERCISES 21-6-2014

(a) Reduce Q to diagonal form by means of an orthonormal basis (that is:

find an orthonormal basis B of V

4

and real numbers λ

i

, i = 1, 2, 3, 4 such that Q(X) = P

4

i=1

λ

i

x

02

, where x

0i

are the components of X with respect to the basis B.

(b) Is Q a positive real quadratic form (this mean that Q(X) ≥ 0 for all X ∈ V

4

and equality holds if and only if X = O).

(c) Compute the maximum and minimum value Q on the unite sphere S

3

. Compute the points of maximum and the points of minimum.

Ex. 0.7. Reduce the following real quadratic forms Q to diagonal form.

Establish if they are positive, negative or what. Find explicitly (if any) non- zero vectors X such that Q(X) > 0 or Q(X) < 0 or Q(X) = 0. Find the maximum and minimum value of Q on the unit sphere and find the points of maximum and the points of minimum.

(a) Q(x, y) = x

2

− 3xy + y

2

.

(b) Q(x, y, z) = −2x

2

− 5y

2

+ 12yz + 7z

2

. (c) Q(x, y, z) = 4xy + 3y

2

+ z

2

.

(d) Q(x, y, z) = 25x

2

− 7y

2

+ 48yz + 7z

2

. (e) Q(x, y, z) = √

2(x

2

+ y

2

+ z

2

) + 2x(y − z) (f ) Q(x, y, z) = 5x

2

− y

2

+ z

2

+ 4xy + 6xz (g) Q(x, y, z) = 2(xy + yz + zx)

Ex. 0.8. Let Q(x

1

, x

2

, x

3

, x

4

, x

5

) = P

5

1=1

x

2i

+ P

i<j

x

i

x

j

. Is Q positive?

Ex. 0.9. Reduce the equations of teh following conics to canonical form, find the symmetry axes, the center, the foci.. (everything in coordinates with respect to the canonical basis).

(a) 5x

2

+ 5y

2

− 6xy + 16 √

2x + 38 = 0.

(b) 5x

2

− 8xy + 5y

2

+ 18x − 18y + 9 = 0 (c) 2x

2

− 2 √

3xy + 2x + 2 √

3y − 5 = 0 (d) x

2

+ 2xy + y

2

+ 2x − 2y = 0

(e) 5x

2

+ 8xy + 5y

2

− 18x − 18y + 9 = 0

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