Maximilian Weigand 17. April 2019
Andreas Hense übung in Anwesenheit
Petra Friederichs InvMod SS 2021
Solve the following exercises using the JupyterHub. Submit a running Python 3 Jupyther Notebook (.ipynb),
and (optionally) a PDF of a fully executed Jupyter notebook to the corresponding eCampus exercise. Please refer
to the eCampus exercise for the deadline.
Total number of points: 25.0
1 Vector and Matrix calculus (19.0 Points)
Matrix multiplication for two matrices AMxN and BNxM is defined as
(AB)μ,λ =
N∑
i=1
aμ,ibi,λ = cμ,λ
This relationship can be simplified vor vectors, as vectors can be seen as matrices with dimension 1 in the second
dimension:
vM = vMx1, → vT1,M
The L2-norm for a vector is defined as
||v|| = ||v||2 =
√√√√ M∑
i=1
x2i
Note also that the dot product of vectors can be represented as a matrix multiplication
u · v = uTv
1.1 Matrix-vector product (Points: 1)
Reformulate the Matrix-Matrix product for a Matrix-vector product between AMxN and vN .
1.2 Vector Norm (Points: 1.5)
Using the definition of the vector norm, show the validity of the equation
||v||2 = vTv, v ∈ RM
Start by treating v and vT as matrices.
1.3 Show (AB)T = BTAT (Points: 2)
Use only the equation for the matrix-matrix product and the fact that the transposed matrix AT of A can be
denoted element-wise by ai,j = aj,i.
A ∈ RM,N , B ∈ RN,P
1.4 Show ||Wm|| =mTWTWm (Points: 2)
1
1.5 Determine the type and size of the following computation results (Points: 2)
Using A ∈ RR,Q,B ∈ RQ,S ,m ∈ RR
Am =
AB =
vTv =
vT =
1.6 Matrix calculus (Points: 2)
Certain derivative operations can be nicely simplified by employing matrix calculus. When certain identities are
used this gravely simplifies certain computations that are central to deriving some inversion problems.
More information on matrix calculus, including a comprehensive list of identities, can be found here:
https://en.wikipedia.org/wiki/Matrix_calculus
m ∈ RN (1)
s
m =
