矩阵乘积的意义

Matrix multiplication

Matrix multiplies vector

Column vector

\[\begin{bmatrix} 1&2&1\\ 2&1&3\\ 1&0&2 \end{bmatrix} \begin{bmatrix}1\\2\\0\end{bmatrix} = 1\begin{bmatrix}1\\2\\1\end{bmatrix}+2\begin{bmatrix}2\\1\\0\end{bmatrix}+0\begin{bmatrix}1\\3\\2\end{bmatrix} = \begin{bmatrix}5\\4\\1\end{bmatrix}.\]

• Notes: A matrix multiplies a column vector generate a new column vector, which is a combination of the column vectors in the matrix.

Row vector

\[\begin{bmatrix}1&2&0\end{bmatrix} \begin{bmatrix} 1&2&1\\ 2&1&3\\ 1&0&2 \end{bmatrix} =1\begin{bmatrix}1&2&1\end{bmatrix}+2\begin{bmatrix}2&1&3\end{bmatrix}+0\begin{bmatrix}1&0&2\end{bmatrix}=\begin{bmatrix}5&4&7\end{bmatrix}.\]

• Notes: A matrix multiplies a row vector generate a new row vector, which is a combination of the row vectors in the matrix.

Matrix multiplies matrix

First situation:

\[\begin{bmatrix} 1&2&1\\ 1&0&-1\\ \end{bmatrix} A_{3 \times n} = \begin{bmatrix} row_1\\ row_2\\ \end{bmatrix} A_{3 \times n} = \begin{bmatrix} row_1 A_{3 \times n}\\ row_2 A_{3 \times n}\\ \end{bmatrix} = B_{2 \times n}.\]

• The problem is transformed to several calculations of multiplication of matrix and row vector.
• The new matrix \(B\) has \(2\) rows, which are the combinations of the row vectors in the matrix \(A\).

Second situation:

\[A_{m \times 3} \begin{bmatrix} 1&1\\ 2&0\\ 1&-1\\ \end{bmatrix} = A_{m \times 3} \begin{bmatrix} column_1&column_2 \end{bmatrix} =\begin{bmatrix} A_{m \times 3}column_1 & A_{m \times 3} column_2 \end{bmatrix} = B_{m \times 2}.\]

• The problem is transformed to several calculations of multiplication of matrix and column vector.
• The new matrix \(B\) has \(2\) columns, which are the combinations of the column vectors in the matrix \(A\).