Let e be the 1 ×m row vector all of whose entries are equal...

Let $\mathrm{e}$ be the $1 \times m$ row vector all of whose entries are equal to 1 . (a) Show that if $A$ is an $m \times n$ matrix, then the $i^{\text {th }}$ entry of the product $\mathbf{v}=\mathbf{e} A$ is the $j^{\text {th }}$ column sum of $A$, meaning the sum of all the entries in its $j^{\text {th }}$ row. (b) Let $W$ denote the $m \times m$ matrix whose diagonal entries are equal to $\frac{1-m}{m}$ and whose off-diagonal entries are all equal to $\frac{1}{m}$. Prove that the column sums of $B=W A$ are all zero. (c) Check both results when $A=\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 1 & 3 \\ -4 & 5 & -1\end{array}\right)$. Remark. If the rows of $A$ represent experimental data values, then the entries of $\frac{1}{m}$ e $A$ represent the means or averages of the data values, while $B=W A$ corresponds to data that has been normalized to have mean 0; see Section 8.8.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where the entries of the product of two matrices are computed as sums of products of corresponding entries from rows of the first matrix and columns of the second. This operation underlies many transformations and computations, including summing entries across rows or columns when one of the matrices is specially structured.

Row Vector of Ones

A row vector whose entries are all one is often used to compute sums along rows or columns when multiplied with another matrix. When such a vector is multiplied by a matrix, each entry in the resulting vector is the sum of the corresponding column of the original matrix, a useful property in statistical computations such as computing means.

Column Sums in Matrices

The column sum of a matrix is the sum of all the entries within a column. This concept is important in many contexts, including statistics and data analysis, as column sums can represent aggregated measures such as total, mean (after appropriate scaling), or serve as checks for balanced transformations of the data.

Data Normalization via Centering Matrices

Data normalization, particularly centering, involves adjusting a dataset so that its mean is zero. This is commonly achieved using a transformation matrix that subtracts the average of the data from each entry. Such a transformation is essential in statistical analysis and machine learning as it can aid in reducing bias and improving the stability of numerical algorithms.

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