Summation of Vectors and Matrices

Definition

Summation of a matrix or vector means adding elements either:

• Column-wise (most common in data analysis)
• Row-wise

A powerful method uses a vector of ones to compute sums and averages using matrix multiplication.

Key Points

• A matrix ( $X\_{n \times m}$ ) has:

  • ( $n$ ) rows (observations)
  • ( $m$ ) columns (features)

• To compute column sums:

  $$1_n^T \cdot X$$

  where ( $1_n^T = [1\ 1\ 1\ ...\ 1]$ ) (row vector of size (1 \times n))

• To compute column averages:

  $$\frac{1}{n} \cdot 1_n^T \cdot X$$

• This works because:

  • Multiplying by a vector of ones adds all rows together

Example / Code

Example 1: Matrix Summation and Average

Given:

Column sum:

Average:

Example 2: Using Ones Vector

Average:

General Formula

$\frac{1}{n} 1_n^T X$

Practice Matrix

Average:

Explanation

• The vector of ones acts like a “sum operator”.

• When you multiply:

  • Each column is summed independently

• Dividing by ( $n$ ) gives the mean of each column

This method is widely used in:

• Machine Learning (feature averaging)
• Statistics (mean calculation)
• Data preprocessing

Output (if any)

• Output of summation: row vector $(1 × m)$
• Output of average: row vector $(1 × m)$

Common Mistakes

• ❌ Using wrong dimension for ones vector → Must match number of rows

• ❌ Dividing by wrong value → Always divide by number of rows ( $n$ ), not columns

• ❌ Arithmetic errors in summation → Double-check sums carefully

• ❌ Confusing row-wise vs column-wise operations

Short Exam Notes (very concise revision points)

• Column sum: ( $1_n^T X$ )
• Column mean: ( $\frac{1}{n} 1_n^T X$ )
• ( $1_n$ ): vector of ones
• Output is always $1 × m$
• Used for fast matrix-based averaging