@@ Line 799: / Line 799: @@
 So the primes only match along the main diagonal. There, they could be considered to cancel out. But we see no value to this. We may as well keep those as '''p'''/'''p''' like all the other entries.
-For a full derivation of this, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Generator embedding, mapping: projection matrix]].
 Here's an example (again, of quarter-comma meantone) of a projection matrix with both its amounts and units:
@@ Line 813: / Line 811: @@
 </math>
+In the following subsections, we examine units-only analyses of some RTT objects. For simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Units-only analyses]].
+===Generator embedding, mapping: projection matrix===
+A <math>\small 𝗽</math>/<math>\small 𝗴</math> generator embedding and a <math>\small 𝗴</math>/<math>\small 𝗽</math> mapping combine to make a <math>\small 𝗽</math>/<math>\small 𝗽</math> projection matrix.
+<math>
+\begin{align}
+\begin{array} {c} G \\[4pt]
+\left[\begin{array} {rrr}
+\frac{\color{ForestGreen}\mathsf{p_1}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\end{array} \right]
+\end{array}
+\begin{array} {c} M \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{\color{BurntOrange}\mathsf{g_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{BurntOrange}\mathsf{g_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{BurntOrange}\mathsf{g_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{OrangeRed}\mathsf{g_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{OrangeRed}\mathsf{g_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{OrangeRed}\mathsf{g_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\end{array} \right]
+\end{array}
+&\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} GM \\[4pt]
+\left[ \begin{array} {rrr}
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{ForestGreen}\mathsf{p_1}})
++
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{ForestGreen}\mathsf{p_1}})
+&
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{NavyBlue}\mathsf{p_2}})
++
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{NavyBlue}\mathsf{p_2}})
+&
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{Plum}\mathsf{p_3}})
++
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{Plum}\mathsf{p_3}})
+\\[6pt]
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{ForestGreen}\mathsf{p_1}})
++
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{ForestGreen}\mathsf{p_1}})
+&
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{NavyBlue}\mathsf{p_2}})
++
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{NavyBlue}\mathsf{p_2}})
+&
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{Plum}\mathsf{p_3}})
++
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{Plum}\mathsf{p_3}})
+\\[6pt]
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{ForestGreen}\mathsf{p_1}})
++
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{ForestGreen}\mathsf{p_1}})
+&
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{NavyBlue}\mathsf{p_2}})
++
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{NavyBlue}\mathsf{p_2}})
+&
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
+(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{Plum}\mathsf{p_3}})
++
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
+(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{Plum}\mathsf{p_3}})
+\\[6pt]
+\end{array} \right]
+\end{array}
+\\[20pt] &\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} P \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\end{array} \right]
+\end{array}
+\end{align}
+</math>
+===Just tuning map, generator embedding: generator tuning map===
+Here, a <math>\mathsf{¢}</math>/<math>\small 𝗽</math> just tuning map and a <math>\small 𝗽</math>/<math>\small 𝗴</math> generator embedding combine to make a <math>\mathsf{¢}</math>/<math>\small 𝗴</math> generator tuning map.
+<math>
+\begin{align}
+\begin{array} {c} 𝒋 \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
+\end{array} \right]
+\end{array}
+\begin{array} {c} G \\[4pt]
+\left[\begin{array} {rrr}
+\frac{\color{ForestGreen}\mathsf{p_1}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\end{array} \right]
+\end{array}
+&\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} 𝒋G \\[4pt]
+\left[ \begin{array} {rrr}
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\color{BurntOrange}\mathsf{g_1}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\color{BurntOrange}\mathsf{g_1}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\color{BurntOrange}\mathsf{g_1}})
+&
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\color{OrangeRed}\mathsf{g_2}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\color{OrangeRed}\mathsf{g_2}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\color{OrangeRed}\mathsf{g_2}})
+\\[6pt]
+\end{array} \right]
+\end{array}
+\\ &\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} 𝒈 \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{{\large\mathsf{¢}}}{\color{BurntOrange}\mathsf{g_1}} & \frac{{\large\mathsf{¢}}}{\color{OrangeRed}\mathsf{g_2}}
+\end{array} \right]
+\end{array}
+\end{align}
+</math>
+===Just tuning map, projection matrix: tuning map===
+A <math>\mathsf{¢}</math>/<math>\small 𝗽</math> just tuning map and a <math>\small 𝗽</math>/<math>\small 𝗽</math> projection matrix combine to make a <math>\mathsf{¢}</math>/<math>\small 𝗽</math> tuning map.
+<math>
+\begin{align}
+\begin{array} {c} 𝒋 \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
+\end{array} \right]
+\end{array}
+\begin{array} {c} P \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\end{array} \right]
+\end{array}
+&\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} 𝒋P \\[4pt]
+\left[ \begin{array} {rrr}
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
+(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
++
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
+(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
++
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})
+(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
+&
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
+(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
++
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
+(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
++
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})
+(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
+&
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
+(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{Plum}\mathsf{p_3}}})
++
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
+(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{Plum}\mathsf{p_3}}})
++
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})
+(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{Plum}\mathsf{p_3}}})
+\end{array} \right]
+\end{array}
+\\ &\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} 𝒕 \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
+\end{array} \right]
+\end{array}
+\end{align}
+</math>
+===Just tuning map, projected interval: tempered interval size===
+A <math>\mathsf{¢}</math>/<math>\small 𝗽</math> just tuning map and a prime-count vector representing a projected interval  combine to give the interval's size in <math>\mathsf{¢}</math>.
+<math>
+\begin{align}
+\begin{array} {c} 𝒋 \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
+\end{array} \right]
+\end{array}
+\begin{array} {c} P\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+\color{ForestGreen}\mathsf{p_1} \\[6pt]
+\color{NavyBlue}\mathsf{p_2} \\[6pt]
+\color{Plum}\mathsf{p_3} \\[6pt]
+\end{array} \right]
+\end{array}
+&\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} 𝒋P\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}})
+\end{array} \right]
+\end{array}
+\\ &\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} 𝒕\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+{\large\mathsf{¢}}
+\end{array} \right]
+\end{array}
+\end{align}
+</math>
+===Generator embedding, mapped interval: projected interval===
+A <math>\small 𝗽</math>/<math>\small 𝗴</math> generator embedding and a generator-count vector (units of <math>\small 𝗴</math>) combine to make a prime-count vector (units of <math>\small 𝗽</math>) representing the projected interval.
+<math>
+\begin{align}
+\begin{array} {c} G \\[4pt]
+\left[\begin{array} {rrr}
+\frac{\color{ForestGreen}\mathsf{p_1}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
+\end{array} \right]
+\end{array}
+\begin{array} {c} M\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+\color{BurntOrange}\mathsf{g_1} \\[6pt]
+\color{OrangeRed}\mathsf{g_2} \\[6pt]
+\end{array} \right]
+\end{array}
+&\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} GM\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})(\cancel{\color{BurntOrange}\mathsf{g_1}}) + (\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})(\cancel{\color{OrangeRed}\mathsf{g_2}}) \\[6pt]
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})(\cancel{\color{BurntOrange}\mathsf{g_1}}) + (\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})(\cancel{\color{OrangeRed}\mathsf{g_2}}) \\[6pt]
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})(\cancel{\color{BurntOrange}\mathsf{g_1}}) + (\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})(\cancel{\color{OrangeRed}\mathsf{g_2}}) \\[6pt]
+\end{array} \right]
+\end{array}
+\\[20pt] &\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} P\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+\color{ForestGreen}\mathsf{p_1} \\[6pt]
+\color{NavyBlue}\mathsf{p_2} \\[6pt]
+\color{Plum}\mathsf{p_3} \\[6pt]
+\end{array} \right]
+\end{array}
+\end{align}
+</math>
+===Projection matrix, interval: projected interval===
+A <math>\small 𝗽</math>/<math>\small 𝗽</math> projection matrix and a prime-count vector (units of <math>\small 𝗽</math>) representing an interval combine to make a new prime-count vector (still with units of <math>\small 𝗽</math>) representing the projected interval.
+<math>
+\begin{align}
+\begin{array} {c} P \\[4pt]
+\left[ \begin{array} {rrr}
+\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\end{array} \right]
+\end{array}
+\begin{array} {c} \textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+\color{ForestGreen}\mathsf{p_1} \\[6pt]
+\color{NavyBlue}\mathsf{p_2} \\[6pt]
+\color{Plum}\mathsf{p_3} \\[6pt]
+\end{array} \right]
+\end{array}
+&\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} P\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}}) \\[6pt]
+(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}}) \\[6pt]
+(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}}) \\[6pt]
+\end{array} \right]
+\end{array}
+\\[20pt] &\begin{array}{c}\\[4pt]=\end{array}
+\begin{array} {c} P\textbf{i} \\[4pt]
+\left[ \begin{array} {rrr}
+\color{ForestGreen}\mathsf{p_1} \\[6pt]
+\color{NavyBlue}\mathsf{p_2} \\[6pt]
+\color{Plum}\mathsf{p_3} \\[6pt]
+\end{array} \right]
+\end{array}
+\end{align}
+</math>
 ==Projection properties==

Projection: Difference between revisions