Matrix Algebra: How to Compute Sums

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.

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 web site, 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 X_r1     Σ X_r2     ...     Σ X_rc ] = S

  where

  1 is an r x 1 sum vector, and 1' is its transpose
  X is an r x c matrix
  Σ X_ri 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 X_ic 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.

  1_r' X1_c = Σ X_rc = 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_rc 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 of This Lesson

Problem 1

Consider matrix 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_i1    Σ A_i2    Σ A_i3 ]

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.

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