Apr 16, 2014 · I think the correct term when referring to general vectors is norm, indicated by ∥∥ ‖ ‖. Modulus is the term specifically used for complex numbers (scalars), and reduces to the concept of absolute value when referring to real numbers. When viewing a complex number as a real pair in the complex plane, then modulus corresponds to the (euclidian) norm on R2 R 2. …

The operator norm is a matrix/operator norm associated with a vector norm. It is defined as ||A||OP =supx≠0 |Ax|n |x| and different for each vector norm. In case of the Euclidian norm |x|2 the operator norm is equivalent to the 2-matrix norm (the maximum singular value, as you already stated). So every vector norm has an associated operator norm, for which sometimes …

Feb 24, 2016 · The norm of a complex vector a a → is not a ⋅a − −−−√ a → ⋅ a →, but a ¯¯¯ ⋅a − −−−√ a → ¯ ⋅ a →. So you should get

What norm do you use? | | A | | 2 = tr(AtA)? If so, just use linearity of the trace functional and the product rule.

Feb 6, 2021 · You ask about the L1 and L2 norms. The L1 norm is the sum of the absolute value of the entries in the vector. The L2 norm is the square root of the sum of the squares of entries of the vector. In general, the Lp norm is the pth root of the sum of …

For example, in matlab, norm (A,2) gives you induced 2-norm, which they simply call the 2-norm. So in that sense, the answer to your question is that the (induced) matrix 2-norm is ≤ ≤ than Frobenius norm, and the two are only equal when all of …

Nov 14, 2011 · Straightforward question, so if it is applied to every element of a vector that means that every one of them is scaled down exactly length times. How did people come up with this, to make it exactl...

There are various definitions of distance. The one you used is the Euclidean distance, which is the square root of the sum of the squares of the components. Your computation is correct, and it is the Euclidean distance from the origin to (7, 2)T (7, 2) T.

The norm gives the length of a a vector as a real number (see def. e.g. here). I further understand that all normed spaces are metric spaces (for a norm induces a metric) but not the other way around (please correct me if I am wrong). Here I am only talking about vector spaces. As an example lets talk about Euclidean distance and Euclidean norm.

Jul 7, 2014 · The Euclidean norm is a special case of this (take p = 2); the taxicab norm is also a special case (take p = 1). Suppose | xi | ≥ | xj | for all 1 ≤ j ≤ n. What happens if p gets really large? Well we would see that the | xi | term would dominate the sum and asymptotically, all of the others would be inconsequential (due to the function being very convex). So what we could say is that ...