- #1

pinodk

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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