Matrix Product (Conventional)/Examples/Column Matrix All 0 except for One 1

From ProofWiki
Jump to navigation Jump to search

Example of (Conventional) Matrix Product

Let $\mathbf A$ be a matrix of order $m \times n$.

For $1 \le i \le n$, let $\mathbf e_i$ be the column matrix of order $n \times 1$ defined as:

$e_k = \delta_{k i}$

where:

$e_k$ is the element of $\mathbf e_i$ whose indices are $\tuple {k, 1}$
$\delta_{k i}$ denotes the Kronecker delta.

Then $\mathbf A \mathbf e_i$ is the column matrix which is equal to the $i$th column of $\mathbf A$.


Proof

Let $\mathbf B = \mathbf A \mathbf e_i$

By definition of matrix product, $\mathbf B$ is a matrix of order $m \times 1$.

Let $b_j$ denote the element of $\mathbf B$ whose indices are $\tuple {j, 1}$.

We have:

$b_j = \displaystyle \sum_{k \mathop = 1}^n a_{j k} e_k$
\(\ds b_j\) \(=\) \(\ds \sum_{k \mathop = 1}^n a_{j k} e_k\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n a_{j k} \delta_{k i}\) Definition of $\mathbf e_i$
\(\ds \) \(=\) \(\ds a_{j i}\)

Hence the result.

$\blacksquare$


Sources