1.7 Centering Matrix

The \(n\times n\) centering matrix is defined by \[H = I - n^{-1}J = I - \frac {1}{n}\underline {1}\,\underline {1}'\] It has the following properties:

1.: \begin {align*} H' & = \big (I - n^{-1}J\big )' = \big (I - n^{-1}\underline {1}\,\underline {1}'\\ & = I' - \frac {1}{n}\big (\underline {1}\,\underline {1}'\big )'\\ & = I - \frac {1}{n}\big (\underline {1}'\big )' \underline {1}'\\ & = I - \frac {1}{n}\underline {1}\,\underline {1}'\\ & = H\\ \end {align*}
2.: \(\displaystyle {H^2 = H}\)
3.: \(\displaystyle {H\underline {1} = \underline {0}}\)
4.: \(\displaystyle {H\underline {X}=\underline {X}-\overline {X}\underline {X}, \overline {X}=\frac {X_1 + X_2 + \cdots + X_n}{n}}\)
5.: \(\displaystyle {\underline {X}'H\underline {X} = \sum \big (X_i - \overline {X}\big )^2}\)

Typical Examples

1.

Determinates

\(A = \begin {pmatrix} 1 & 5 & 2 & 1\\ 3 & 7 & 4 & 5\\ 2 & 9 & 1 & 2\\ 4 & 0 & 1 & 3\\ \end {pmatrix}\)

\begin {align*} \begin {vmatrix} A\\ \end {vmatrix} & = \begin {vmatrix} 1 & 5 & 2 & 1\\ 3 & 7 & 4 & 5\\ 2 & 9 & 1 & 2\\ 4 & 0 & 1 & 3\\ \end {vmatrix} = \begin {matrix} C_1 - 4C_3\\ \longrightarrow \\ \end {matrix}\quad \begin {vmatrix} -7 & 5 & 2 & 1\\ -13 & 7 & 4 & 5\\ -2 & 9 & 1 & 2\\ 0 & 0 & 1 & 3\\ \end {vmatrix}\\\\ & =\begin {matrix} C_4 - 3C_3\\ \longrightarrow \\ \end {matrix}\quad \begin {vmatrix} -7 & 5 & 2 & -5 \\ -13 & 7 & 4 & -7\\ -2 & 9 & 1 & -1\\ 0 & 0 & 1 & 0\\ \end {vmatrix} = \begin {matrix} C_1 - 2C_4\\ \longrightarrow \\ C_2 + 9C_4\\ \end {matrix}\quad \begin {vmatrix} 3 & -40 & 2 & -5\\ 1 & -56 & 4 & -7\\ 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 0\\ \end {vmatrix}\\\\ & = \begin {vmatrix} 3 & -40\\ 1 & -56\\ \end {vmatrix} \begin {vmatrix} 1 & -1\\ 1 & 0\\ \end {vmatrix}\\\\ & = (-168 + 40) (0+1)\\ & = -128\\ \end {align*}

2.

\(H_3 = I_3 -3^{-1}J_3\) \begin {align*} H_3 & = I_3 - 3^{-1} J_3\\\\ & = \begin {pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end {pmatrix}-\frac {1}{3}\begin {pmatrix} 1\\ 1\\ 1\\ \end {pmatrix}\begin {pmatrix} 1 & 1 & 1\\ \end {pmatrix}\\\\ & = \begin {pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end {pmatrix}-\frac {1}{3}\begin {pmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1\\ \end {pmatrix}\\\\ & = \frac {1}{3}\begin {pmatrix} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2\\ \end {pmatrix}\\\\ \end {align*}

1.

\(H'=H\) clear

2.

\(H^2 = H\) \begin {align*} H^2 & = \Bigg [\frac {1}{3}\begin {pmatrix} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2\\ \end {pmatrix}\Bigg ]\Bigg [\frac {1}{3}\begin {pmatrix} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2\\ \end {pmatrix}\Bigg ]\\\\ & = \frac {1}{9}\begin {pmatrix} 6 & -3 & -3 \\ -3 & 6 & -3\\ -3 & -3 & 6\\ \end {pmatrix}= \frac {1}{3}\begin {pmatrix} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2\\ \end {pmatrix}\\\\ & = H\\ \end {align*}

3.

\(H\underline {1} = \underline {0}\) \begin {align*} H\underline {1} & = \frac {1}{3}\begin {pmatrix} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2\\ \end {pmatrix} \begin {pmatrix} 1\\ 1\\ 1\\ \end {pmatrix}= \frac {1}{3}\begin {pmatrix} 2-1-1\\ -1+2-1\\ -1-1+2\\ \end {pmatrix}\\\\ & = \frac {1}{3}\begin {pmatrix} 0\\ 0\\ 0\\ \end {pmatrix}\\\\ & = \underline {0}\\ \end {align*}

4.

\(H\underline {X} = \underline {X} - \overline {X}\underline {1}\) \begin {align*} H\underline {X} & = \frac {1}{3}\begin {pmatrix} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2\\ \end {pmatrix}\begin {pmatrix} X_1 \\ X_2\\ X_3\\ \end {pmatrix}= \frac {1}{3}\begin {pmatrix} 2X_1-X_2-X_3\\ -X_1+2X_2-X_3\\ -X_1-X_2+2X_3\\ \end {pmatrix}\\\\ & = \frac {1}{3}\begin {pmatrix} 3X_1 - X_1 - X_2 - X_3\\ 3X_2 - X_1 - X_2 - X_3\\ 3X_3 - X_1 - X_2 - X_3\\ \end {pmatrix}\\\\ & = \frac {1}{3}\begin {pmatrix} 3X_1\\ 3X_2\\ 3X_3\\ \end {pmatrix}-\frac {1}{3}\begin {pmatrix} X_1 + X_2 + X_3\\ X_1 + X_2 + X_3\\ X_1 + X_2 + X_3\\ \end {pmatrix}\\\\ & = \begin {pmatrix} X_1\\ X_2\\ X_3\\ \end {pmatrix} - \overline {X}\begin {pmatrix} 1\\ 1\\ 1\\ \end {pmatrix}\\\\ & = \underline {X}-\overline {X}\underline {1}\\ \end {align*}

5.

\(\underline {X}'H\underline {X} = \sum \big (X_i-\overline {X}\big )^2\)

\[H\underline {X} = \begin {pmatrix} X_1 - \overline {X}\\ X_2 - \overline {X}\\ X_3 - \overline {X}\\ \end {pmatrix} \]

\begin {align*} \underline {X}'H\underline {X} & = \underline {X}' H^2 \underline {X}\\\\ & = \underline {X}'HH\underline {X}\\ & = \underline {X}'H'H\underline {X}\\ & = \big (H\underline {X}\big )'H\underline {X}\\ & = \big (X_1-\overline {X}\big )^2 + \big (X_2-\overline {X}\big )^2 + \big (X_3-\overline {X}\big )^2\\\\ & = \sum (X_i-\overline {X}\big )^2\\\\\\ \end {align*}

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