10.5 Row-Vector-Matrix Multiplication via Dot Products

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 10.5.1 Example of Row-Vector-Matrix Multiplication via Dot Products
Constructions: N/A
Generalizations: N/A

Properties: 10.7 Properties of Row-Vector-Matrix Multiplication
Sufficiencies: N/A
Questions: N/A

Row-Vector-Matrix Multiplication via Dot Products

Let matrix $A \in R^{m \times n}$ and 3.1 Column Vector $\vec{x} \in R^{m \times 1}$ . We define the row-vector-matrix product ${\vec{x}}^{T} = \vec{b} \in R^{1 \times n}$ via dot products using an entry-by-entry approach where the output is a 3.4 Row Vector. The $k$ th entry of the product $\vec{b} = {\vec{x}}^{T} A$ can be calculated by an inner product between the $k$ th column of $A$ and the vector $\vec{x}$ :

b_{k} = {entry}_{(1, k)} ({\vec{x}}^{T} A) = \vec{x} \cdot (A (:, k)) = \sum_{i = 1}^{m} a_{i k} x_{i}

for column index $k \in {1, 2, \dots, n}$ . When we calculate every entry of the output vector, we produce

{\vec{x}}^{T} A = [\begin{matrix} b_{1} & b_{2} & \dots & b_{n} \end{matrix}]

In other words, we are taking the dot product of our column vector $\vec{x}$ and the $k$ th column of $A$ .

Remark. Recall that when working with matrices and vectors, the inner dimensions must agree.

[\vec{b}]_{1 \times n} = [{\vec{x}}^{T}]_{1 \times m} \cdot [A]_{m \times n}

Number of columns of ${\vec{x}}^{T}$ matches the rows of $A$
Output $\vec{b}$ is defined by the outer dimensions, $1 \times n$
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Remark. Similar to 10.4 Row-Vector-Matrix Multiplication via Linear Combinations, we also partition the output vector into column partitions. Let's look at finding the entries of $\vec{b}$ .

\begin{aligned} \vec{b} & = {\vec{x}}^{T} \cdot A \\ = {\vec{x}}^{T} \cdot [\begin{array}{ccc} A (:, 1) & A (:, 2) & \dots & A (:, n) \end{array}] \\ = [\begin{array}{ccc} b_{1} & b_{2} & \dots & b_{n} \end{array}] \\ ⟹ b_{1} & = {entry}_{(1, 1)} ({\vec{x}}^{T} \cdot A) \\ = \vec{x} \cdot A (:, 1) \\ = [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{array}] \cdot [\begin{array}{c} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{array}] \\ = x_{1} a_{11} + x_{2} a_{21} + \dots + x_{m} a_{m 1} \\ ⟹ b_{2} & = {entry}_{(1, 2)} ({\vec{x}}^{T \cdot} A) \\ = \vec{x} \cdot A (:, 2) \\ = [\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{array}] \cdot [\begin{array}{c} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{array}] \\ = x_{1} a_{12} + x_{2} a_{22} + \dots x_{m} a_{m 2} \\ b_{2} & = \sum_{i = 1}^{m} x_{i} a_{i 2} \end{aligned}

The last entry of output $\vec{b} = {\vec{x}}^{T} \cdot A$ is

\begin{aligned} ⟹ b_{k} & = {entry}_{(1, n)} ({\vec{x}}^{T} \cdot A) \\ = \vec{x} \cdot A (:, n) \\ = {[\begin{array}{c} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{array}]}_{1 \times m} \cdot {[\begin{array}{c} a_{1 k} \\ a_{2 k} \\ ⋮ \\ a_{m n} \end{array}]}_{1 \times m} \\ = x_{1} a_{1 n} + x_{2} a_{2 n} + \dots + x_{m} a_{m n} \end{aligned}

Putting this all together, we have the entry-by-entry definition.

\begin{aligned} [\begin{array}{ccc} b_{1} & b_{2} & \dots & b_{n} \end{array}]_{1 \times n} & = [\begin{array}{ccc} x_{1} & x_{2} & \dots & x_{m} \end{array}]_{1 \times m} {[\begin{array}{ccc} a_{11} & a_{12} & a_{1 m} \\ a_{21} & a_{22} & \dots & a_{2 m} \\ ⋮ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & a_{m n} \end{array}]}_{m \times n} \\ = [\begin{array}{ccc} x_{1} a_{11} + x_{2} a_{21} + \dots + x_{m} a_{m 1} & \dots & x_{1} a_{1 n} + x_{2} a_{2 n} + \dots + x_{m} a_{m n} \end{array}] \end{aligned}

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Remark. If we factor out the scalar multiples, we can produce the 10.4 Row-Vector-Matrix Multiplication via Linear Combinations definition.

\begin{aligned} ⟹ [\vec{b}]_{1 \times m} & = x_{1} [\begin{array}{c} a_{11} & a_{12} & \dots & a_{1 n} \end{array}] \\ + x_{2} [\begin{array}{c} a_{21} & a_{22} & \dots & a_{2 n} \end{array}] \\ ⋮ \\ + x_{m} [\begin{array}{c} a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}] \end{aligned}

Therefore, both methods produce the exact same output.

Dot product version focuses on producing scalar data
Linear combination version focuses on producing vector data
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