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   =   
1
2
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 columns, 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.

    X1   =       
    Σ X1c
    Σ X2c
    . . .
    Σ Xrc
    		   =    S

    where

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

  • Matrix-wide. And finally, it is possible to compute a grand sum of all the elements in matrix X, as shown below.

    1r' X1c = Σ Xrc = S

    where

    1r is an r x 1 sum vector, and 1r' is its transpose
    1c is an c x 1 sum vector
    X is an r x c matrix
    Σ Xrc 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   =   
3 5 1
9 1 4

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 Ai1    Σ Ai2    Σ Ai3 ]

Hint: Use the sum vector, 12.

Solution

The 1 x 3 vector b' can be derived by premultiplying matrix A by the transpose of 12, as shown below.

b'  =  
1 1
 
3 5 1
9 1 4
12' A

b'  =  
3 + 9   5 + 1   1 + 4

b'  =  
12   6   5

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