Matrix L1-Norm: Max of Column Sums Explained

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In summary, the conversation discusses a homework problem involving an m x n matrix A and the l1-norm of the matrix. The speaker mentions trying to use the definition of the vector l1-norm with Ax for some x, where the l1-norm of x is 1, but this results in the l1-norm of A equaling the sum of the row sums of A. The speaker asks for help in identifying where they made a mistake and requests to see someone else's work for reference.
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gucci1
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Hi, for a homework problem I was asked to show that for an m x n matrix A, the l1-norm of the matrix is the max of the sums of the columns. What I tried to do was plug into the definition of the vector l1-norm with Ax for some x st the l1-norm of x is 1. This shows, after a bit of simplifying, the l1-norm of A is equal to the sum of the row sums of A. This is not at all what I was supposed to show, so does anyone know where I made a mistake? Thanks for any help!
 
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gucci said:
so does anyone know where I made a mistake? Thanks for any help!

Better if you show your work. At any case, have a look here.
 

FAQ: Matrix L1-Norm: Max of Column Sums Explained

What is the Matrix L1-Norm?

The Matrix L1-Norm is a mathematical concept used to measure the magnitude of a matrix. It is also known as the maximum absolute column sum or the Taxicab norm. It is calculated by taking the sum of the absolute values of each column in a matrix and then selecting the maximum value from those sums.

How is the Matrix L1-Norm useful?

The Matrix L1-Norm is useful in many fields, including statistics, machine learning, and signal processing. It can be used to compare the size of different matrices, identify outliers, and determine the most important features in a dataset.

What does "Max of Column Sums" mean in the context of Matrix L1-Norm?

In the context of Matrix L1-Norm, "Max of Column Sums" refers to the process of finding the maximum value after taking the sum of the absolute values of each column in a matrix. This value represents the Matrix L1-Norm of the matrix.

How is Matrix L1-Norm different from other matrix norms?

Matrix L1-Norm is different from other matrix norms, such as the L2-Norm or the Frobenius norm, because it only considers the absolute values of the elements in a matrix. This makes it more robust to outliers and can provide a more accurate measure of the magnitude of a matrix.

Can the Matrix L1-Norm be used for non-square matrices?

Yes, the Matrix L1-Norm can be used for non-square matrices. It can be applied to any matrix with any number of rows and columns as long as the values are real numbers. However, it is important to note that the Matrix L1-Norm may not always be defined for non-square matrices as it depends on the dimensions of the matrix and the values of its elements.

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