Visualizing norms as a unit circle
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Asked 7 years, 7 months ago. Active 2 years, 3 months ago.
Viewed 38k times. If anyone can explain to me what is wrong with my reasoning here, I would appreciate it! Alt 2, 3 3 gold badges 15 15 silver badges 34 34 bronze badges. Kristian Kristian 1, 1 1 gold badge 15 15 silver badges 28 28 bronze badges. Perhaps this is what I've overlooked. But is there a way to find out the vector which will give the supremum value?
Active Oldest Votes. Will Jagy Will Jagy k 5 5 gold badges silver badges bronze badges. But how do you know this? I must admit that this theory is very new to me, so it hasn't quite sunk in yet. I will look for the proof, so I also can see this!In mathematicsa matrix norm is a vector norm in a vector space whose elements vectors are matrices of given dimensions.
A matrix norm that satisfies this additional property is called a sub-multiplicative norm in some books, the terminology matrix norm is used only for those norms which are sub-multiplicative. For symmetric or hermitian Awe have equality in 1 for the 2-norm, since in this case the 2-norm is precisely the spectral radius of A.
For an arbitrary matrix, we may not have equality for any norm; a counterexample being given by. In any case, for square matrices we have the spectral radius formula :. This inequality can be derived from the fact that the trace of a matrix is equal to the sum of its eigenvalues.
It can be shown to be equivalent to the above definitions using the Cauchy—Schwarz inequality. This is a different norm from the induced p -norm see above and the Schatten p -norm see belowbut the notation is the same. It is used in robust data analysis and sparse coding.
This norm can be defined in various ways:. Recall that the trace function returns the sum of diagonal entries of a square matrix.
The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using Cauchy—Schwarz inequality. This norm is not sub-multiplicative. The Schatten p -norms arise when applying the p -norm to the vector of singular values of a matrix. These norms again share the notation with the induced and entrywise p -norms, but they are different. All Schatten norms are sub-multiplicative.
All induced norms are consistent by definition. Induced norms are compatible with the inducing vector norm by definition. From Wikipedia, the free encyclopedia. Norm on a vector space of matrices. Main article: Operator norm. Main article: Hilbert—Schmidt operator.
See also: Frobenius inner product. Further information: Schatten norm.In mathematicsthe operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c.
Thus the image of a bounded set under a continuous operator is also bounded.
Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of Ait then seems natural to take the infimum of the numbers c such that the above inequality holds for all v in V. In other words, we measure the "size" of A by how much it "lengthens" vectors in the "biggest" case.
So we define the operator norm of A as. The infimum is attained as the set of all such c is closednonemptyand bounded from below. It is important to bear in mind that this operator norm depends on the choice of norms for the normed vector spaces V and W. Every real m -by- n matrix corresponds to a linear map from R n to R m. Each pair of the plethora of vector norms applicable to real vector spaces induces an operator norm for all m -by- n matrices of real numbers; these induced norms form a subset of matrix norms.
This can be viewed as an infinite-dimensional analogue of the Euclidean space C n. The operator norm is indeed a norm on the space of all bounded operators between V and W.
This means. For bounded operators on Vthis implies that operator multiplication is jointly continuous. It follows from the definition that a sequence of operators converge in operator norm means they converge uniformly on bounded sets.
Some common operator norms are easy to calculate, and others are NP-hard. The norm of the adjoint or transpose can be computed as follows. Suppose H is a real or complex Hilbert space. In general, the spectral radius of A is bounded above by the operator norm of A :. To see why equality may not always hold, consider the Jordan canonical form of a matrix in the finite-dimensional case. Because there are non-zero entries on the superdiagonal, equality may be violated. The quasinilpotent operators is one class of such examples.
However, when a matrix N is normalits Jordan canonical form is diagonal up to unitary equivalence ; this is the spectral theorem. In that case it is easy to see that. The spectral theorem can be extended to normal operators in general. Therefore, the above equality holds for any bounded normal operator N.
The space of bounded operators on Hwith the topology induced by operator norm, is not separable. For example, consider the Hilbert space L 2 [0,1].
This implies the space of bounded operators on L 2 [0,1] is not separable, in operator norm.An matrix can be considered as a particular kind of vectorand its norm is any function that maps to a real number that satisfies the following required properties: Positivity:. In addition to the three required properties for matrix norm, some of them also satisfy these additional properties not required of all matrix norms: Subordinance:.
We now consider some commonly used matrix norms. Element-wise norms If we treat the elements of are the elements of an -dimensional vector, then the p-norm of this vector can be used as the p-norm of :. The Frobenius norm of a unitary orthogonal if real matrix satisfying or is:.
Induced or operator norms of a matrix is based on any vector norm. Note that the norm of the identity matrix is. We now prove the matrix norm defined above satisfy all properties given above. Recall. Specifically, the matrix p-norm can be based on the vector p-normas defined in the following for. Whenis maximum absolute column sum:. Proof: The 1-norm of vector iswe have.
Whenis maximum absolute row sum:. Proof: Whenis normalized if. The norm of vector is:. Whenis the spectral norm, the greatest singular value ofwhich is the square root of the greatest eigenvalue ofi. Proof: When, and we have. The right-hand side of the equation above is a weighted average of the eigenvalueswhich is maximized if they are weighted by a normalized vector withby which the greatest eigenvalue is maximally weighted while all others are weighted by 0.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. The fact that such rescalings exist follows from the fact that norms on a finite-dimensional space are pairwise equivalent. The point of this is that there are a lot of norms on the space of matrices if we don't make any additional requirements on them. Is this the kind of answer you were looking for?
Or do you want the two norms to have additional properties? Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Submultiplicative matrix norm: Max Norm Ask Question. Asked 9 years, 10 months ago. Active 1 year, 10 months ago. Viewed 11k times. Active Oldest Votes. Ian Morris Ian Morris 5, 2 2 gold badges 24 24 silver badges 57 57 bronze badges. Furthermore, can you refer me to any book on this?
In the absence of any other recommendations, you might try the books referred to in the Wikipedia article on matrix norms linked to by Wadim above. In particular, the book by Horn and Johnson seems to be widely appreciated. Mustafa Said Mustafa Said 2, 1 1 gold badge 22 22 silver badges 24 24 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. The Overflow Blog.
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The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Proof for infinity matrix norm Ask Question. Asked 3 years, 1 month ago. Active 3 years, 1 month ago. Viewed 6k times. Active Oldest Votes.
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Formally the -norm of is defined as:. A p-th-root of a summation of all elements to the p-th power is what we call a norm. Hereby we are going to look into some of these norms in details. The first norm we are going to discuss is a -norm. By definition, -norm of is. Strictly speaking, -norm is not actually a norm. It is a cardinality function which has its definition in the form of -norm, though many people call it a norm. It is a bit tricky to work with because there is a presence of zeroth-power and zeroth-root in it.
Obviously any will become one, but the problems of the definition of zeroth-power and especially zeroth-root is messing things around here. So in reality, most mathematicians and engineers use this definition of -norm instead:. Because it is a number of non-zero element, there is so many applications that use -norm.
The sparsest solution means the solution which has fewest non-zero entries, i. This problem is usually regarding as a optimisation problem of -norm or -optimisation.
A standard minimisation problem is formulated as:. However, doing so is not an easy task. In many case, -minimisation problem is relaxed to be higher-order norm problem such as -minimisation and -minimisation. Following the definition of norm, -norm of is defined as.
If the -norm is computed for a difference between two vectors or matrices, that is. The most popular of all norm is the -norm. It is used in almost every field of engineering and science as a whole. Following the basic definition, -norm is defined as. As in -norm, if the Euclidean norm is computed for a vector difference, it is known as a Euclidean distance :. MSE is. Because of this, we will now discuss about the optimisation of. As in -optimisation case, the problem of minimising -norm is formulated by.
Assume that the constraint matrix has full rank, this problem is now a underdertermined system which has infinite solutions. The goal in this case is to draw out the best solution, i. This could be a very tedious work if it was to be computed directly. Luckily it is a mathematical trick that can help us a lot in this work. Take derivative of this equation equal to zero to find a optimal solution and get. By using this equation, we can now instantly compute an optimal solution of the -optimisation problem.
This equation is well known as the Moore-Penrose Pseudoinverse and the problem itself is usually known as Least Square problem, Least Square regression, or Least Square optimisation.