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and determine the sig- nature of the quadratic form q(x1e1+x2e2+x3e3)=x1x2+x2x3

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MA 1112: Linear Algebra II Tutorial problems, March 26, 2019

1.Compute the eigenvalues of the matrix

0 1 0 1 0 1 0 1 0

, and determine the sig- nature of the quadratic form

q(x1e1+x2e2+x3e3)=x1x2+x2x3. 2.Let

ϕ(x1,x2)=sin2(x1−x2)−e12(x12+x22)+2c x1x2.

Furthermore, letAbe the symmetric 2×2-matrix with entriesai j= ∂

2ϕ

∂xi∂xj(0, 0, 0).

(a) Write down the matrixA.

(b) Determine all values of the parameter c for which the corresponding quadratic form is positive definite.

(c) Doesϕhave a local minimum at the origin (0, 0) forc= −3/5?

3.For the scalar product (A,B)=tr(ABT) on the space of alln×n-matrices, show that

(AB,AB)É(A,A)(B,B)

for all matrices AandB. (Hint: write (AB,AB) explicitly using the entries of A andB, and use the Cauchy–Schwartz inequality).

Optional question: Show that inRn, it is impossible to findn+2 vectors that only form obtuse angles (that is, (vi,vj)<0 for alli 6=j).

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