What is the definition of the Euclidean norm?

sqrt(sum(x^2))

What is the definition of the Cauchy-Schwarz inequality?

|dot prodict of u and v| ≤ ||u|| * ||v||

How can you prove divergence of a vector sequence?

Show that one sequence of the vector sequence is divergent

Do vector sequences have fixed dimensions?

Yes, all elements of the sequence have a fixed dimension

If a vector sequence is convergent, what does this also imply?

iff the limit of the difference of the sequence and the limit itself is zero

iff the limit of the norm of the difference of the sequence and the limit itself is zero (this works for the 1-norm, 2-norm and the infinity norm)

How do the 1-,2- and infinity norms relate to each other?

The 2-norm is always greater equal the infinity norm

The 1-norm is always greater equal the 2-norm

What is a property of a norm concerning the values it returns?

A norm only returns non-negative numbers

For which norms does the Cauchy-Schwarz inequality hold?

It holds for the 1-norm and 2-norm but not for the infinity norm

Considering the unit balls of the 1-, 2-, and infinity norms in two- and three dimensional space, which one of them is the biggest/smallest

1-norm is the smallest

infinity norm is the biggest

Are the 1-, 2-, and infinity norms all continuous?

Yes. They are all continuous functions

Is infinity included as a possible accumulation point in a multivariate space?

No it is not, since there is no proper definition for it

The composition of continuous functions is also continuous for m = 1, does this also hold for compositions in higher dimensions?

No. It does not, since we don’t have a suitable definition of a product of vectors in that case

It only holds for additive composition

What condition must be fulfilled such that the directional derivative of f, equals the dot product of the gradient of f and the corresponding direction?

f must be continuously partial differentiable

This also holds for any task related to steepest ascent/descent

Does partial differentiability imply continuity?

No. It does not

Does continuity imply partial differentiability

What condition must be fulfilled such that the second order derivatives of different variables are the same?

It must be twice continuously differentiable

This also holds for any task concerning the Hessian matrix

This also holds for any task concerning minima/maxima on open domains

If f has global extremum at a point, does this imply that it has also has a local extremum at that point?

Yes. It does

What must be fulfilled for (semi) negative and (semi) positive definiteness?

A must be symmetric

In what direction does a function decrease the most?

It decreases the most in the direction of the negative gradient

Is the identity matrix positive definite?

Yes. It is

Last changeda year ago