Stat Trek

Teach yourself statistics

Stat Trek

Teach yourself statistics

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.

    1rX1c = 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.