- #1
pinodk
- 21
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I have a couple of questions on this matter...
QUESTION 1.)
I am to show a certain Lemma, that the definition of a matrix norm
||A|| = max(||x||<=1) ||Ax||
obides with the definitions of a standard vector norm
a.) ||A||>=0 for all A and ||A||=0 <=> A=0
b.) ||tA|| = |t| ||A|| for any real number t
c.) ||A+B|| <=||A||+||B|| for all A,B
Ive shown b and c, although b is confusing me somewhat:
b.) ||tA|| must satisfy the following
||tA|| = max(||x||<=1) ||tAx||
= |t| max(||x||<=1) ||Ax||
= |t| ||A||
I thinkthis proves it, but still I am not quite sure what would happen if t is negative from the beginning.
Could someone tell me if ||tA|| is then equal to |t| ||A|| and why?
The really important part of of question 1 is getting a hint of how to show a.), since i have no idea how to...
Question 2
Show that
||A|| = max(||x||!=0) ||Ax||/||x||
From a definition in the book, I know that ||Ax||<=||A|| ||x||
then i must have that
||A|| <= max(||x||!=0) ||A|| ||x||/||x||
giving me that
||A|| <= max(||x||!=0) ||A||
which proves it.
Is this proof ok?
Question 3
The definition i use above is proven in the book, the first line of this proof says
||Ax|| = ||A (x/||x||) x||
why is this so?
Is it cause x/||x|| is a unit vector, and thus makes no difference when multiplied on A?
Anyone who can help me, will receive an appropriate e-hug :-)
Cheers
Daniel
QUESTION 1.)
I am to show a certain Lemma, that the definition of a matrix norm
||A|| = max(||x||<=1) ||Ax||
obides with the definitions of a standard vector norm
a.) ||A||>=0 for all A and ||A||=0 <=> A=0
b.) ||tA|| = |t| ||A|| for any real number t
c.) ||A+B|| <=||A||+||B|| for all A,B
Ive shown b and c, although b is confusing me somewhat:
b.) ||tA|| must satisfy the following
||tA|| = max(||x||<=1) ||tAx||
= |t| max(||x||<=1) ||Ax||
= |t| ||A||
I thinkthis proves it, but still I am not quite sure what would happen if t is negative from the beginning.
Could someone tell me if ||tA|| is then equal to |t| ||A|| and why?
The really important part of of question 1 is getting a hint of how to show a.), since i have no idea how to...
Question 2
Show that
||A|| = max(||x||!=0) ||Ax||/||x||
From a definition in the book, I know that ||Ax||<=||A|| ||x||
then i must have that
||A|| <= max(||x||!=0) ||A|| ||x||/||x||
giving me that
||A|| <= max(||x||!=0) ||A||
which proves it.
Is this proof ok?
Question 3
The definition i use above is proven in the book, the first line of this proof says
||Ax|| = ||A (x/||x||) x||
why is this so?
Is it cause x/||x|| is a unit vector, and thus makes no difference when multiplied on A?
Anyone who can help me, will receive an appropriate e-hug :-)
Cheers
Daniel