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Mathematics LibreTexts

14.1: Properties of the Standard Basis

The standard notion of the length of a vector \(x=(x_{1},x_{2},\ldots,x_{n})\) \(\in\) \(\mathbb{R}^{n}\) is 
$$||x||=\sqrt{x\cdot x}=\sqrt{(x_{1})^{2}+(x_{2})^{2}+\cdots(x_{n})^{2}}\, .$$
The canonical/standard basis in \(\mathbb{R}^{n}\)

$$e_{1}=\begin{pmatrix}1\\0\\ \vdots \\ 0\end{pmatrix}, e_{2}=\begin{pmatrix}0\\1\\ \vdots \\ 0\end{pmatrix}, \ldots, e_{n}=\begin{pmatrix}0\\0\\ \vdots \\ 1\end{pmatrix}\,$$
has many useful properties with respect to the dot product and lengths:

1.    Each of the standard basis vectors has unit length:
\[
\|e_{i}\|=\sqrt{e_{i}\cdot e_{i}}=\sqrt{e_{i}^{T}e_{i}}=1\, .\]
2.    The standard basis vectors are \(\textit{orthogonal}\) (in other words, at right angles or perpendicular):  
\[
e_{i}\cdot e_{j} = e_{i}^{T}e_{j}=0 \textit{ when } i\neq j
\]

This is summarized by
\[
e_{i}^{T}e_{j}=\delta_{ij}=\left\{ \begin{array}{cc}
1 & \qquad i=j \\
0 & \qquad i\neq j \\
\end{array}\right. ,
\]
where \(\delta_{ij}\) is the \(\textit{Kronecker delta}\).  Notice that the Kronecker delta gives the entries of the identity matrix.

Given column vectors \(v\) and \(w\), we have seen that the dot product \(v\cdot w\) is the same as the matrix multiplication \(v^{T}w\).  This is an \(\textit{inner product}\) on \(\Re^{n}\).  We can also form the \(\textit{outer product}\) \(vw^{T}\), which gives a square matrix.
The outer product on the standard basis vectors is interesting.  Set 

$$\Pi_{1} = e_{1}e_{1}^{T} = \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0\end{pmatrix} \begin{pmatrix} 1 & 0 & \cdots & 0\end{pmatrix}$$

$$~~~~~~= \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & 0\end{pmatrix}$$

$$\vdots$$

$$\Pi_{n} = e_{n}e_{n}^{T} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1\end{pmatrix} \begin{pmatrix} 0 & 0 & \cdots & 1\end{pmatrix}$$

$$~~~~~~= \begin{pmatrix} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & 1\end{pmatrix}$$

In short, \(\Pi_{i}\) is the diagonal square matrix with a \(1\) in the \(i\)th diagonal position and zeros everywhere else.

Notice that 
\(\Pi_{i}\Pi_{j}=e_{i}e_{i}^{T}e_{j}e_{j}^{T}=e_{i}\delta_{ij}e_{j}^{T}\).  Then:
\[
\Pi_{i}\Pi_{j} = \left\{ \begin{array}{cc}
\Pi_{i} & \qquad i=j \\
0 & \qquad i\neq j \\
\end{array}\right. .
\]
Moreover, for a diagonal matrix \(D\) with diagonal entries \(\lambda_{1},\ldots, \lambda_{n}\), we can write
\[
D= \lambda_{1}\Pi_{1} + \cdots + \lambda_{n}\Pi_{n}.
\]

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