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 = |
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x = |
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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 XWithin 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 XMatrix-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 = |
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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' = |
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12' | A |
b' = |
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b' = |
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Notice that each element of vector b' is indeed equal to the sum of column elements from matrix A.