Exercise No. 3 – Matrix Trinity

Mathematical Foundations of Computer Vision

Michael Breuß Released:18.11.2011

Assigned to:Tutorial at 24.11.2011

Assignment 4 – Matrix Reloaded

Exercise No. 1 – Enter the Matrix

We consider a functiony=ϕ(x)withy∈IR^m,x∈IRⁿ, and whereϕis some transformation.

TheJacobian matrixofϕis defined as

∂y

∂x =







∂y1

∂x1 · · · _∂x^∂y1 .. n

. . .. ...

∂y_m

∂x1 · · · ^∂y_∂x^m

n





∈IR^m×n (1)

(a) Lety=Axwithy∈IR^m,x∈IRⁿ, and whereA= (aij),A∈IR^m×n, does not depend onx.

Prove or disprove: ^∂y_∂x =A (2pts)

(b) Letf =x^>Axbe given wheref ∈IR,x∈IRⁿ,A= (aij),A∈IR^n×n.

Compute ^∂f_∂x for(i)Anot symmetric, and for(ii)Asymmetric. (4pts) (c) Letf(z) =y^>(z)x(z)wherez∈IRⁿ,x(z)∈IRⁿ,y(z)∈IRⁿ.

Compute ^∂f_∂z. (2pts)

(d) Letϕ(x) =kx−vk₂, wherex, v∈IRⁿ.

Compute ^∂ϕ_∂x. (2pts)

Exercise No. 2 – Differentiate the Matrix

Now, letA= (aij)be am×nmatrix withaij =aij(t),t∈IR. Then

d

dtA(t) = A(t) =˙







da₁₁

dt · · · ^da_dt¹ⁿ ... . .. ...

dam1

dt · · · ^da_dt^mn





∈IR^m×n (2)

(a) LetB, C ∈IR^n×nwithB = (b_ij),b_ij =b_ij(t)andC= (c_ij),c_ij=c_ij(t).

LetBC=I. Compute the equation resulting out of d

dt[BC] = d dt[I]

(4pts)

(b) LetA= (a_ij)∈IR^m×nbe invertible, witha_ij =a_ij(t).

Compute _dt^d A⁻¹

. (4pts)

1

Exercise No. 3 – Matrix Trinity

We consider again thesimilarityproperty.

We remember, that for given A ∈ IR^n×n there is a similar matrix Λ if we have an orthogonal matrix Q∈IR^n×nwithA=QΛQ^>.

(a) Prove that the following implication holds:

IfA can be made similar to a diagonal matrixΛ,thenAis symmetric. (4pts) We have made use already of such matricesQcomposed of the eigenvectors ofA. We give this some more basement:

(b) Prove that the following assertion holds:

For a symmetric matrixA, the eigenvalues are real. (4pts)

We supplement this by:

(c) Prove that the following assertion holds:

For a symmetric matrixA, the eigenvectors to different eigenvalues are orthogonal. (4pts)

2