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Matrix scaling

a.k.a.

• Matrix-scalar multiplication
• Matrix-scalar product
• Scalar multiplication
• Scaling

Define

… in terms of scalar multiplication as the operation

\[\begin{equation*}
  c A
  = c
  \begin{pmatrix}
    \vphantom{\bigg(}
    \vec{a}_1 & \vec{a}_2 & \cdots
  \end{pmatrix}
  =
  \begin{pmatrix}
    \vphantom{\bigg(}
    c \vec{a}_1 & c \vec{a}_2 & \cdots
  \end{pmatrix},
\end{equation*}
\]

where	\(c, A\)	are the factors and
	\(cA\)	is the product.

Discuss

• Multiplicative identity

  … is \(1\), so

  \[\begin{equation*}
    1 A = A.
  \end{equation*}
  \]

• Commutative property

  … holds, so

  \[\begin{equation*}
    c A = A c.
  \end{equation*}
  \]

• Associative property

  … holds, so

  \[\begin{equation*}
    c (d A) = (c d) A.
  \end{equation*}
  \]

• Distributive property

  … holds over matrix addition, where

  \[\begin{equation*}
    c (A + B) = c A + c B.
  \end{equation*}
  \]