GeoLing12: The product of matrices. C ¯ ontents: • Multiplying

GeoLing12: The product of matrices. C ¯ ontents: • Multiplying

Geometria Lingotto.
GeoLing12: The product of matrices.
Contents:
¯
• Multiplying matrices.
Recommended lecture: Leling 9.
¯
EXERCISES
1. Compute the following products:

(a)


1 2 0 1 1 ·



(c)


0 −1 −1 1 1 ·


2
3
0
−3
5
1
3
4
−2
1




 ; (b)




4 3 3 1 1 ·








0 −1 −1 1 1


 ; (d)  4 3


3
1
1
·




1 2
0 1 1
2
3
4
−2
1

(a)
N=
0 1
0 0
(b)
0
 0
N =
 0
0
1
0
0
0

0
0 

1 
0
0
0
0
0
3. Calculate N 3 = N · N · N :

(a)
0 1 0
N = 0 0 1 
0 0 0
Ingegneria dell’Autoveicolo, GL12


1
(b)
0
 0
N =
 0
0
1
0
0
0
0
1
0
0





2
2
1
3
3
3
0
4
4
−3 −2 −2
5
1
1
2. Calculate N 2 = N · N :


0
0 

1 
0
Geometria






Geometria Lingotto.
4. Is there a matrix X such that A · X = B ?

(a)
A=
1 2 0

1

1
(b) A =
1

1

1
(c) A =
1
2
2
2
0
2
2

1
 3 



1 1 ,
B=
 4 
 −2 
1



0
1 2
0 ,
B= 1 2 
0
1 0

 
0
1


0
1 
,
B=
−1
1
5. Compute A · At :
(a)
A=
0 1
−1 0
(b)
A=
1 3
2 −1
(c)
A=
cos(θ) − sin(θ)
sin(θ) cos(θ)
6. Prove that A · At is a symmetric matrix.
7. Take A ∈ R2,2 and prove that if A · At = 0 than A = 0. Is the same true for
A ∈ Z2,2
2 ?.
8. Is it true that ρ(A · B) = min{ρ(A), ρ(B)} ?.
9. Matrix multiplication according to Strassen.
Check that
a11 a12
a21 a22
b11 b12
c11 c12
·
=
b21 b22
c21 c22

c11



c12
, dove
c21



c22
= p1 + p4 − p5 + p 7 ,
= p3 + p5 ,
= p2 + p4 ,
= p1 + p3 − p2 + p 6 .

p1 = (a11 + a22 )(b11 + b22 ),




p2 = (a21 + a22 )b11 ,




 p3 = a11 (b12 − b22 ),
1 2
p4 = a22 (b21 − b11 ),
and
. Multiply according to Strassen
·
3 4


p5 = (a11 + a12 )b22 ,




p = (a21 − a11 )(b11 + b12 ),


 6
p7 = (a12 − a22 )(b21 + b22 ).
Ingegneria dell’Autoveicolo, GL12
2
Geometria
Geometria Lingotto.
5 6
7 8
10. Complex numbers as matrices. Let J =
0 −1
1 0
.
(a) Check that J is a square root of −1, i.e. J 2 = −1, with −1 = −I = −I2 .
0
x −y
x −y 0
0
0
0
(b) Let z = xI + yJ =
and z = x I + y J =
. Calculate
y x
y 0 x0
z · z0 .
(c) Verify z t = xI − yJ and z · z t = (x2 + y 2 )I .





0 1 0 0
0
0 1 0
0
−1 0 0 0 
0


0 0 1


0
11. Quaternions. Let i = 
 0 0 0 −1 , j = −1 0 0 0 , k =  0
0 0 1 0
0 −1 0 0
−1
Verify the following relations, discovered by Hamilton, hold:

0 0 1
0 −1 0
.
1 0 0
0 0 0
i2 = j 2 = k 2 = ijk = −1 .
These formulas are engraved on one of Dublin’s bridges, cf. http://en.wikipedia.org/wiki/Quaternion
A quaternion q is a matrix:


a
b
c
d
 −b a −d c 
=
q=
 −c d
a −b
−d −c b
a

1
0
= a
0
0
0
1
0
0
0
0
1
0
 
0
0


0 −1
+b
0  0
0
1
1
0
0
0
 
0
0
0 0


0
0 0 0
+c
0 −1 −1 0
0 −1
1 0
1
0
0
0
 
0
0


1  0
+d
0  0
0
−1
Check that a = 0 and b2 + c2 + d2 = 1 imply q 2 = −1.
Ingegneria dell’Autoveicolo, GL12
3
Geometria

0 0 1
0 −1 0
.
1 0 0
0 0 0