9.5 Scalar-Matrix Multiplication

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 9.5.1 Matrix as Linear Combination of Outer Products
Constructions: N/A
Generalizations: 9.4 Matrix-Matrix Addition

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Scalar-Matrix Multiplication

Let matrix $A \in R^{m \times n}$ and scalar $α \in R$ . The scalar-matrix product $α \cdot A$ is obtained by multiplying each entry of $A$ by the scalar $α$ .

{scalar-matrix product}_{i k} = A_{i k} \cdot α

Remark. The $\cdot$ symbol is overused in linear algebra. To confirm that we are indeed doing scalar-matrix multiplication, we should check the dimensions of our operands.

α_{1 \times 1} \cdot A_{m \times n}

& Recall that all scalars are of size $1 \times 1$ .
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Remark. The $\cdot$ symbol is quite powerful because we are actually doing $m \times n$ operations using just $1$ symbol!

α \cdot A = α \cdot [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}] = [\begin{matrix} α \cdot a_{11} & α \cdot a_{12} & \dots & α \cdot a_{1 n} \\ α \cdot a_{21} & α \cdot a_{22} & \dots & α \cdot a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ α \cdot a_{m 1} & α \cdot a_{m 2} & \dots & α \cdot a_{m n} \end{matrix}]

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