Matrix norm and Matrix measure

In summary, the conversation discusses the properties of matrix measures and the convexity of the measure function. The measure function, u(A), is shown to be convex for any matrix A, including a Hessian matrix derived from a nonconvex function. The definition of convexity and properties (M2) and (M5) are mentioned as key factors in proving the convexity of the measure. The discussion also suggests exploring the role of norms in understanding the convexity of the measure function. Another question is raised about finding the maximum value of the measure function, which can be achieved by looking at bound points in the (u,x,y) space. However, it is emphasized that a careful mathematical proof of convexity is necessary to fully understand
  • #1
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Hi, initially I would like to share this link: https://books.google.com.tr/books?id=gWeVPoBmBZ8C&pg=PA25&lpg=PA25&dq=matrix+measure+properties&source=bl&ots=N1unizFvG6&sig=kxijoOVlPAacZDEdyyCwam4RQnQ&hl=en&sa=X&ved=2ahUKEwjd7o-Ap53dAhWJGuwKHdRbAO04ChDoATABegQICBAB#v=onepage&q=matrix measure properties&f=false. Here, you are allowed to view the pages between 22-26 and they are about MEASURE MATRİX.

According to those pages, matrix measure u(A) is a convex function, and A is a matrix form. So no matter which matrix we put into "u()" function, it always ensures convexity. For instance as you can see on page 26, $$u_1(A)= max_j(a_{jj}+\sum_{i=!j} |a_{ij}|)$$ where a_ij are elements of matrix A. And let^s say that A is Hessian matrix which is derived from very complicated nonconvex nonlinear function.

MY QUESTION is: Although our Hessian matrix's elements are nonlinear and nonconvex, HOW is it POSSIBLE that the measure of Hessian matrix is convex? I can not believe measure of matrix is convex because for previous Hessian matrix, all elements of it which also take place in "u()" function are nonconvex nonlinear and nonconvex. Could you explain this situation to me?
 
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  • #2
Check the definition of convexity. Then look at properties (M2) and (M5) on page 22 -- this meets the definition of convexity. In general homogeneity under (positive) rescaling and sub-additivity will do it. You need to go through these details slowly and verify this for yourself.

edit:

what worries me is this:

mertcan said:
MY QUESTION is: Although our Hessian matrix's elements are nonlinear and nonconvex, HOW is it POSSIBLE that the measure of Hessian matrix is convex?

which suggests you don't know what convex means or aren't grasping that it is referring to the the 'measure' which is convex by design.

Your post title mentions norms and measure. If it were me, I'd look at the norms in detail and explore why they are convex. (Hint: M2 and M5 still apply.) Induced Norms (and Schatten Norms) are quite common and are what these 'matrix measures' are being built on...
 
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  • #3
StoneTemplePython said:
Check the definition of convexity. Then look at properties (M2) and (M5) on page 22 -- this meets the definition of convexity. In general homogeneity under (positive) rescaling and sub-additivity will do it. You need to go through these details slowly and verify this for yourself.

edit:

what worries me is this:
which suggests you don't know what convex means or aren't grasping that it is referring to the the 'measure' which is convex by design.

Your post title mentions norms and measure. If it were me, I'd look at the norms in detail and explore why they are convex. (Hint: M2 and M5 still apply.) Induced Norms (and Schatten Norms) are quite common and are what these 'matrix measures' are being built on...
Thanks for return @StoneTemplePython , I would like to ask another question as a reply to my post 1. Let's say we have 2 dimensions (x,y) for the elements of Hessian matrix in total, and concentrating on 5<=x<=10 and -3<=y<=2. Also we know that measure function is convex a function which means u(A) is convex. Besides, we can imagine (u,x,y) space and u is a function of x,y(because elements of Hessian consists of x,y). Here, can we say that in order to obtain MAXIMUM value of measure function for the intervals 5<=x<=10 and -3<=y<=2 we can just look at bound points? (for instance for a basic parabola function(z=x'^2+y^2), maximum value of parabola for intervals 5<=x<=10 and -3<=y<=2 would be at bounds...)
 
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  • #4
mertcan said:
Thanks for return @StoneTemplePython , I would like to ask another question as a reply to my post 1. Let's say we have 2 dimensions (x,y) for the elements of Hessian matrix in total, and concentrating on 5<=x<=10 and -3<=y<=2. Also we know that measure function is convex a function which means u(A) is convex. Besides, we can imagine (u,x,y) space and u is a function of x,y(because elements of Hessian consists of x,y). Here, can we say that in order to obtain MAXIMUM value of measure function for the intervals 5<=x<=10 and -3<=y<=2 we can just look at bound points? (for instance for a basic parabola function(z=x'^2+y^2), maximum value of parabola for intervals 5<=x<=10 and -3<=y<=2 would be at bounds...)

Again I'd suggest working with norms first... these 'bound points' you talk about are (x,y) coordinates that get mapped via a function (and really its second derivative) to the Hessian and then you apply a convex norm to it. I still think you need to write out what exactly is convex here and why. I.e. a careful mathematical writeup and proof of the convexity.

The domain for your convex function is given by the coordinates of the Hessian, not x and y. Alternatively you can have a composition of functions applied to these x and y coordinates, but you don't have per se reason to believe that the composition is convex.
 

1. What is a matrix norm?

A matrix norm is a mathematical concept used to measure the size or magnitude of a matrix. It is defined as a function that takes a matrix as its input and outputs a non-negative value. Matrix norms are used to quantify how much a matrix distorts the space in which it operates.

2. What is the difference between a matrix norm and a matrix measure?

A matrix norm and a matrix measure are two closely related concepts. While a matrix norm measures the size or magnitude of a matrix, a matrix measure measures the complexity or structure of a matrix. In other words, a matrix norm gives a single number that represents the size of a matrix, while a matrix measure gives a more detailed description of the matrix.

3. How is a matrix norm calculated?

The calculation of a matrix norm depends on its definition. There are many different types of matrix norms, such as the Frobenius norm, the spectral norm, and the maximum norm. Each norm has its own formula for calculation, but they all share the same basic principles of measuring the size of a matrix.

4. What are some applications of matrix norms and measures?

Matrix norms and measures have various applications in mathematics, engineering, and computer science. They are used to analyze and solve systems of linear equations, calculate eigenvalues and eigenvectors, and determine the stability of dynamical systems. They are also used in data analysis, signal processing, and machine learning.

5. Can a matrix norm be negative?

No, a matrix norm is always a non-negative value. By definition, a norm must be non-negative, so it cannot be negative. However, individual elements of a matrix can be negative, which may affect the value of the norm. But the norm itself will always be a positive number.

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