Homework Statement
##P=A(A^*A)^{-1}A^*##
where A is a mxn real/complex matrix and ##A^*A## is invertible.
##A^*## means the conjugate transpose of A.
Homework EquationsThe Attempt at a Solution
Let y be in the range(A), such that
##y = Ax## for some ##x##.
We can see that ##PA =...
Hmmm. I might be missing something here:
So if I square both sides I obtain:
##||x||_1= (\sum |x_i|)^2##
and
##||x||_2=(\sum |x_i|^2)##
Since, the square of sums of absolute values is always greater or equal to the sum of squares, we have proved that ##||x||_1 \geq ||x||_2##.
Is what I just...
Homework Statement
Show that ##||x||_1\geq ||x||_2##
2. Homework Equations
##||x||_1 = \sum_{i=1}^n |x_i|##
##||x||_2 = (\sum_{i=1}^n |x_i|^2)^.5##
The Attempt at a Solution
I am having a hard time with this, because the question just seems so trivial, that I don't even know how to prove...
The induced matrix norm for a square matrix ##A## is defined as:
##\lVert A \rVert= sup\frac{\lVert Ax \rVert}{\lVert x \rVert}##
where ##\lVert x \rVert## is a vector norm.
sup = supremum
My question is: is the numerator ##\lVert Ax \rVert## a vector norm?
Thanks for the answer.
To find out if a matrix is orthogonal (I know there are various ways), is it sufficient to show that the dot product of any given 2 column vectors in the vector is zero?
Hello. I have a question regarding curvature and second derivatives. I have always been confused regarding what is concave/convex and what corresponds to negative/positive curvature, negative/positive second derivative.
If we consider the profile shown in the following picture...
When looking at Elliptic PDEs that describe a physical system, do these typically not involve a time term?
I have yet to see an elliptic PDE involving a time term, which seem to be associated with parabolic/hyperbolic PDEs rather than elliptic.
Can anyone confirm?
But even if the viscosity varies with temperature, it doesn't necessarily mean it is a non-Newtonian fluid right?
As long as the viscosity does not vary with the strain rate, the shear stress is still a linear function of the strain rate, even if the viscosity is varying with other parameters?
I was reading this:
http://www.creatis.insa-lyon.fr/~dsarrut/bib/Archive/others/phys/www.mas.ncl.ac.uk/%257Esbrooks/book/nish.mit.edu/2006/Textbook/Nodes/chap06/node29.html
Under the first figure it states "Figure 6.20: The boundary layer at a stagnation point on an airfoil has a constant...