SLIDE 1
Lecture6.1: Whatwewillnot betalkingabout
Optimization and Computational Linear Algebra for Data ScienceLéo Miolane
Warning
1/6/ Homeworks 1 exams
.Lecture6.1: Whatwewillnot betalkingabout Optimization and - - PowerPoint PPT Presentation
Lecture6.1: Whatwewillnot betalkingabout Optimization and - - PowerPoint PPT Presentation
Lecture6.1: Whatwewillnot betalkingabout Optimization and Computational Linear Algebra for Data Science Lo Miolane Warning / Home works 1 exams . 1/6 The determinant There exists a function det : R n n R called the determinant
SLIDE 2
SLIDE 3
The determinant
2/6There exists a function det : Rn×n → R called the determinant that verifies det(M) = 0 ⇐ ⇒ M is not invertible. The determinant can be computed using the following formula: det(M) =
ÿ
σ∈Sn‘(‡)
nŸ
i=1Mi,σ(i) =
=⇒
som- ver all
rrorqdeu
:L
" -y -
- Maru, Mano
- than
,
numbers
1,2 -
- n
g
- 1
Exe :
n=4
,2341
is
a
- ak Yatra
)
depending
ah t .
amodeling of
t .
. - - 4 SLIDE 4
Geometrical interpretation
3/6£2
A=f¥?dD)
/detlA=ad-bI
Vz
- A
I detest
= lad- bet
- ff
del
- CA) , o
⇒
A
- o
⇐s
Va , vz
lin
.dep
.⇐
A is not invertible
SLIDE 5
Link with eigenvalues
4/6X
is
aneigenvalue
- f A
rot Kala - did) ⇒
there exists
vto
such
that Ao
.⇒
Kala
- AID) f {o}
⇐s A
- XII
is
not
invertible
.⇐
det ( A
- kid)
O
- function
- f
X
that
we write SLIDE 6
The characteristic polynomial
5/6- Pala
)
is
a
polynomial
in
se .Ed: Let's
consider
A = (ta z)
Pala)
=det CA
- aid)
det ( FIFA)
=( n -a)( 2-a)
- 2 =lnZ-3at1T
- Pa
is called
the
characteristic polynomial of A
its
roots
are
the
eigenvalues of A
.- deg CPA) f n
hence
A
has at
most
n distincteigenvalues
. SLIDE 7
Example
6/6 I 1Let's
take A=fy ;
- Ro - Tsing
- ÷:) ;
for D= Tk
i 'Pala)
=det CA
- aid)
det ( IIe)
= Ia2tI- For all
aER
,Pala
) = htt 71 70
Hence
A-
does
not have
any real eigenvalues
.- PACE)
a.Ztt
=C-1) t 1
= 0i
is
a
complex eigenvalue
- f A
(
- iheeisothae)