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Relation Between Matrix-Vector Multiplication and Matrix Partitions

Recall the different methods of doing matrix-vector multiplication. These are related to the different partitions of a matrix.

8.13 Column Partition of a Matrix
- 10.1 Matrix-Column-Vector Multiplication via Linear Combinations
- 10.2 Matrix-Column-Vector Multiplication via Dot Products
8.14 Row Partition of a Matrix
- 10.4 Row-Vector-Matrix Multiplication via Linear Combinations
- 10.5 Row-Vector-Matrix Multiplication via Dot Products

Below is a graphic illustrating the relationships.

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Text Elements

scalarized version

(data broken into
individual entries)
Matrix-Row-Vector
Multiplication
vectorized version

(data chunked into
vector-sized pieces)
Matrix-Column-Vector
Multiplication

Embedded files

8a9f80f541b321794f4fbec2aa21051b8b369096: $A \cdot \vec{x} = \vec{b}$
349d20c01f1a6ab00d324e2a24cc4a73821c82e6: $A \cdot \vec{x} = \vec{b}$
3e06897a5bcc06638c3bbd111a63bdb567acd4ce: $\vec{b} = \sum_{i = 1}^{m} x_{i} A (i, :)$
52edc1c69e12a2be439256dd86297f5d7485dad2: $\vec{b} = \sum_{k = 1}^{n} x_{k} A (:, k)$
c5a456f1fdd62c390b2cb8e5d19140e22588a04a: $b_{i} = [A (i, :)]^{T} \cdot \vec{x}$
83e11f75cea9f244c233ffae4c8d4a19b28238fe: $= \sum_{k = 1}^{n} a_{i k} x_{k}$
a7977a63a2b20179c5351d1c4672ee07a9705369: $b_{k} = \vec{x} \cdot A (:, k)$
fa620616aafcbc35868c5b0e2d6b71b6fa51d28f: $= \sum_{i = 1}^{m} x_{i} a_{i k}$
c0958aeb128d030012bb43043ad2ae2160041e06: ${\vec{x}}^{T} \cdot A = \vec{b}$
c110adcb1bf0c6e31a02a0fba332ce3d63afdbee: ${\vec{x}}^{T} \cdot A = \vec{b}$

Notice:

The modeling matrix is on the left for matrix-column-vector multiplication but is on the right for matrix-row-vector multiplication.
The linear combination methods vectorize the data while the dot product methods focus on scalarized data

When solving real world problems that require us to decide which method to use, two useful questions to ask are:

Is the modeling matrix the left or right factor?
Do we want vectorized or scalarized data?

Remark. The relation between the methods will prove useful when solving computer science problems.
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