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seen Sep 30 at 3:10

Apr
27
accepted How is it shown that a Hermitian matrix will be positive definite?
Apr
26
comment How is it shown that a Hermitian matrix will be positive definite?
Thank you very much, this is the sort of thing I was looking for. So a matrix is positive definite if it is hermitian, finite dimensional(is that what all that $k\leq n$ stuff means?), and the determinant is positive? Other question was about the hint, I tried it and got: $z = \left( \begin{array}{c} |i|^2 \\ -(1)(-i) \end{array}\right) \Rightarrow \overline{z}^{t}A_{1}z = 4$, did I misunderstand the inputs?
Apr
26
comment How is it shown that a Hermitian matrix will be positive definite?
@J.M.: sorry, but I still can't understand your idea... First, I thought I had already worked out the $m=1$ case, right? Second, I don't see where, how, or why to make $a,b,c,d$ evaluate to something manifestly nonpositive... Also when you say quadratic form, you are referring to a specific type of mapping, or just any polynomial with terms of degree 2?
Apr
26
comment How is it shown that a Hermitian matrix will be positive definite?
@J.M. : do I need to show that $a^2 +b^2 +c^2 +d^2 \gt ad -bc$? Do you know how to do this? (Sorry if this is trivial...)
Apr
26
comment How is it shown that a Hermitian matrix will be positive definite?
thank you for this answer, it is good to know that this is a possibility, unfortunately I think this method is still a little too advanced.
Apr
26
comment How is it shown that a Hermitian matrix will be positive definite?
@J.M: as to the definition of positive definite: I think it has so far just been $\alpha$ is positive definite $:\Leftrightarrow \forall v \in V, v\neq 0 : \alpha(v,v) \gt 0$... so pretty basic and we haven't covered how eigenvalues relate to that yet...
Apr
26
comment How is it shown that a Hermitian matrix will be positive definite?
@J.M. : To your previous comment, thank you for the tip! Does something like this look like the right direction? $z = (a+bi, c+di)^t \Rightarrow m(a^2 + b^2) -2(ad-bc) +m(c^2+d^2)$ and since $m(a^2 + b^2) -2(ad-bc) +m(c^2+d^2) \geq 0 \Leftrightarrow m \geq \frac{2(ad-bc)}{a^2 + b^2 +c^2 +d^2}$
Apr
26
asked How is it shown that a Hermitian matrix will be positive definite?
Apr
23
accepted What is the difference between finding a basis for a complex and a real space?
Apr
23
comment What is the difference between finding a basis for a complex and a real space?
Thank you very much!
Apr
23
asked What is the difference between finding a basis for a complex and a real space?
Apr
21
revised Why does $a_n = (1+\frac{2}{n})^{n}$ converge to $e^2$?
deleted 1 characters in body
Apr
17
revised How to show certain things related to scalar products
removed unnecessary ">" caused by blockquote formatting
Apr
17
accepted How to show certain things related to scalar products
Apr
17
comment How to show certain things related to scalar products
thank you for this answer
Apr
17
comment How to show certain things related to scalar products
thanks for all your help
Apr
17
comment How to show certain things related to scalar products
Thank you for this answer. I think I should be able to manage (3) now. With the geometric interpretation I am still a bit stuck. When you mention those 4 corners of the parallelogram, is each one an ordered pair, like $u= (u_1,u_2)$? Also, is "Pythagoras" written in the problem as a pointer to use the Pythagorean theorem somehow? I just don't know exactly where to apply it, I try to 'extend' the parallelogram horizontally to form right angles, but then the lengths become unclear...
Apr
17
revised How to show certain things related to scalar products
added 143 characters in body
Apr
17
comment How to show certain things related to scalar products
@Theo: yeah that was the 'workaround' I was trying out, but now I can go ahead and replace the alphas thanks to what you just taught me.
Apr
17
revised How to show certain things related to scalar products
added 2 characters in body