Summary Sum of Matrix Elements

How to Compute Sums of Matrix Elements
This lesson explains how to use matrix methods to compute sums of vector elements and sums of matrix elements.


How to Compute Sums: Vector Elements
The sum vector 1n is a 1 x n column vector having all n elements equal to one. The main use of the sum vector is to find the sum of the
elements from another 1 x n vector, say vector xn.

Let's demonstrate with an example.

1 1
1 = 1 x = 2
1 3

Then, the sum of elements from vector x is:

Σ xi = 1'x = ( 1 * 1 ) + ( 1 * 2) + ( 1 * 3 ) = 1 + 2 + 3 = 6

Note: For this website, we have defined the sum vector to be a column vector. In other places, you may see it defined as a row vector.


How to Compute Sums: Matrix Elements
The sum vector is also used to find the sum of matrix elements. Matrix elements can be summed in three different ways: within column
within rows, and matrix-wide.



Within columns. Probably, the most frequent application is to sum elements within columns, as shown below.

1'X = [ Σ Xr1 Σ Xr2 ... Σ Xrc ] = S

where

1 is an r x 1 sum vector, and 1' is its transpose
X is an r x c matrix
Σ Xri is the sum of elements from column i of matrix X
S is a 1 x c row matrix whose elements are column sums from matrix X



Within rows. It is also possible to sum elements within rows, as shown below.

Σ X1c

Σ X2c
X1 = = S
...
Σ Xrc


where

1 is an c x 1 sum vector
X is an r x c matrix