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# 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

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.