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 1_{n} 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 x_{n}.
Let's demonstrate with an example.
1 = 

x = 

Then, the sum of elements from vector x is:
Σ x_{i} = 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 columns, within rows, and matrixwide.
Within columns. Probably, the most frequent application is to sum elements within columns, as shown below.
1'X = [ Σ X_{r}_{1} Σ X_{r}_{2} ... Σ X_{r}_{c} ] = S
where
1 is an r x 1 sum vector, and 1' is its transpose
X is an r x c matrix
Σ X_{r}_{i} 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 XWithin rows. It is also possible to sum elements within rows, as shown below.
X1 = Σ X_{1}_{c} Σ X_{2}_{c} . . . Σ X_{r}_{c} = S where
1 is an c x 1 sum vector
X is an r x c matrix
Σ X_{i}_{c} is the sum of elements from row i of matrix X
S is a r x 1 column matrix whose elements are row sums from matrix XMatrixwide. And finally, it is possible to compute a grand sum of all the elements in matrix X, as shown below.
1_{r}' X1_{c} = Σ X_{r}_{c} = S
where
1_{r} is an r x 1 sum vector, and 1_{r}' is its transpose
1_{c} is an c x 1 sum vector
X is an r x c matrix
Σ X_{r}_{c} is the sum of all elements from matrix X
S is a real number equal to the sum of all elements from matrix X
Test Your Understanding
Problem 1
Consider matrix A.
A = 

Using matrix methods, create a 1 x 3 vector b', such that the elements of b' are the sum of column elements from A. That is,
b' = [ Σ A_{i}_{1} Σ A_{i}_{2} Σ A_{i}_{3} ]
Hint: Use the sum vector, 1_{2}.
Solution
The 1 x 3 vector b' can be derived by premultiplying matrix A by the transpose of 1_{2}, as shown below.
b' = 



1_{2}'  A 
b' = 

b' = 

Notice that each element of vector b' is indeed equal to the sum of column elements from matrix A.