Question

Let $A$ be an $m \times n$ matrix that is normalized, meaning that each of its column sums is zero. Show that $A B$, where $B$ is any $n \times k$ matrix, is also normalized.

Name: Problem 6, Chapter 8: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2023-04-11T01:12:45-08:00
Channel: Victor Salazar
Description: Let A be an m ×n matrix that is normalized, meaning that each of its column sums is zero. Show that A B, where B is any n ×k matrix, is also normalized.

    Let $A$ be an $m \times n$ matrix that is normalized, meaning that each of its column sums is zero. Show that $A B$, where $B$ is any $n \times k$ matrix, is also normalized.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 8, Problem 6

Instant Answer

Solved by Expert Victor Salazar

04/11/2023

Step 1

A matrix $A$ is said to be normalized if the sum of the elements in each of its columns is zero. Mathematically, this can be expressed as $\sum_{i=1}^m A_{ij} = 0$ for each column $j$ where $1 \leq j \leq n$. Here, $A_{ij}$ represents the element in the $i$-th row Show more…

Show all steps

Thanks for your feedback!

Let $A$ be an $m \times n$ matrix that is normalized, meaning that each of its column sums is zero. Show that $A B$, where $B$ is any $n \times k$ matrix, is also normalized.

Play audio

Feedback

Victor Salazar and 89 other educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Normalized Matrix

A normalized matrix, in this context, refers to a matrix where each column sums to zero. This property implies that the accumulation of all entries in any given column results in zero, which is a specific form of balance or centering in the matrix data.

Matrix Multiplication

Matrix multiplication involves combining two matrices in such a way that each element of the product is formed by the dot product of the corresponding row and column from the multiplied matrices. This operation is linear, which is key in preserving certain properties of the original matrices.

Linearity and Preservation of Column Sums

Because summation is a linear operation, when matrix A (with zero column sums) multiplies another matrix B, the property that the sum of the entries in each column of A is zero is maintained in the product. Essentially, the zeros are ‘carried through’ the multiplication due to the distributive nature of the dot product over addition.

Please give Ace some feedback