How can I learn about the maximum and eigenvalues of matrices?

[azizz]

Does anybody have a good book/website where I can find good information on how to use the maximum on matrices. I have to prove an expression involving the maximum and eigenvalue of matrices. But I don't know how to link those to together. I think I can figure this out, if only I had some good information source :)

Regards, Azizz

 

[morphism]

What do you mean by "the maximum"?

 

[azizz]

Sorry I went to fast here. With the maximum I meant the largest (or maximal) eigenvalue, for example

$$\lambda_{\max}(A) = \max_{\| x \| =1} x^* A x$$

Then my question is: what do I know of this operator? Is it, eg, linear?

 

[morphism]

You mean is the function $\lambda_\max$ linear on the space of matrices? Certainly not.

 

[azizz]

Ok, but I think this holds true:

Suppose A-B is hermitian and positive definite, then

$$\max_{\|x\|=1} x^*(A-B)x \geq x^*(A-B) x = x^*Ax - x^*Bx \leq \lambda_{\max}(A) - \lambda_{\max}(B)$$

 

[azizz]

Found partly what I needed:

$$\lambda_{\max}(A)I \geq A \geq \lambda_{\min}(A)I$$

$$\beta I > A \iff \beta > \lambda_{\max}(A)$$

Now all I have to know is what is known for the eigenvalue of two matrices? That is:

$$\lambda_{\max}(A+B) = ...$$

Is there any expression I can use for such an equality (or perhaps inequility)?

 

[morphism]

I don't think you can say anything intelligent for arbitrary matrices A and B. (But I could be wrong!)