# What is the relationship between $||.||_{max}$ and energy of a matrix?

I was interested to find a relationship between $$||.||_{max}$$, i.e. max-norm of a matrix with that of its energy. $$||.||_{max}$$ of a matrix is defined by the maximum entry in the matrix. Generally, energy of a matrix is defined by the sum of diagonal elements ($$\sigma$$) of the matrix after singular value decomposition. I found, an inequality concerning representation of energy of a non-negative matrix in terms of $$||.||_{max}$$. But, what happens when the constrained of non-negativity is lifted? Is there any generalized relationship?

• Hi, welcome to MSE. I'm sure there's a good question here, but the standards of the site require context (otherwise your question may be closed/voted down). To start with, you should provide the definitions of the $\|\cdot\|_{\max}$ norm and the energy of a matrix, plus the inequalities you're referring to for non-negative matrices. Further, it might be helpful to be more specific about the kind of relationship you're looking for, so that our answerers can be as helpful as possible. – Theo Bendit Feb 19 at 4:39
• Thanks. I edited the question with more details and definition of the operations. Let me know if that helps. Thanks. – God_Help Feb 19 at 5:44
• It's better. So, if I'm interpreting this correctly, the energy is the sum of the singular values (or, equivalently, the energy of $A$ is $\operatorname{trace}\sqrt{A^\top A}$)? What is this inequality that you've got when the entries are non-negative? That is, what specifically are we supposed to generalise? – Theo Bendit Feb 19 at 10:20
• Thanks. I am sharing the paper which I followed to find the relationship between the energy of the matrix and $||.||_{max}$. sciencedirect.com/science/article/pii/S0022247X06003258 In this work, the author has presented a relationship between energy of a non-negative matrix with that of the $||.||_{max}$. But, it would only work for non -ve matrices. I am dealing with negative matrices. Like, an orthogonal matrix generated from DCT basis. Is there any work which would lead to a generalized matrix, whose energy would be calculated in terms of $||,||_{max}$. – God_Help Feb 19 at 21:34