Turn a unit input and watch its score changeOrdinary coordinates show a teal unit input v, its orange output Mv, the blue output component Q(v)v along the input, and a dashed perpendicular remainder. The dashed circle is the set of all unit inputs. Initially v=(3,1)/sqrt(10), so its score is 2.6. A closer view of the same unit input shows its two perpendicular eigenvector components. Their signed coefficients are about 0.447 and 0.894, and their shares of squared length are 20 percent and 80 percent. Turning the input changes its score between 1 and 3. A chart at the bottom plots the score of every direction from -180 to 180 degrees, a wave between 1 and 3 that touches 3 at 45 and -135 degrees (the u2 line) and 1 at -45 and 135 degrees (the u1 line); a dot on the curve marks the current input. The optional resized copy does not change the unit input shown in these plots. Turn the input. Keep its length at 1. Ordinary coordinates in both pictures Input Output Output along input Perpendicular remainder Apply M and project its output Split the same input into pieces Closer view of the input -1 1 v Mv Q(v)v Dashed circle: all unit inputs c1​u1​ c2​u2​ v Signed amount Squared-length share c1​ = 0.447 20% c2​ = 0.894 80% 1 (λ1​) 2 3 (λ2​) Unit-input score: 2.6 The score for every direction at once The dot is the current input; the small dots at ±45° and ±135° are the eigenvector directions λ1​ = 1 λ2​ = 3 −180° −90° 0° 90° 180° 0 1 2 3
Turn a unit input and watch its score changeOrdinary coordinates show a teal unit input v, its orange output Mv, the blue output component Q(v)v along the input, and a dashed perpendicular remainder. The dashed circle is the set of all unit inputs. Initially v=(3,1)/sqrt(10), so its score is 2.6. A closer view of the same unit input shows its two perpendicular eigenvector components. Their signed coefficients are about 0.447 and 0.894, and their shares of squared length are 20 percent and 80 percent. Turning the input changes its score between 1 and 3. A chart at the bottom plots the score of every direction from -180 to 180 degrees, a wave between 1 and 3 that touches 3 at 45 and -135 degrees (the u2 line) and 1 at -45 and 135 degrees (the u1 line); a dot on the curve marks the current input. The optional resized copy does not change the unit input shown in these plots. Turn the input. Keep its length at 1. Ordinary coordinates in both pictures Input Output Output along input Perpendicular remainder -1 1 v Mv Q(v)v Dashed circle: all unit inputs The same input, split into pieces Closer view; ordinary coordinates c1​u1​ c2​u2​ v Signed amount Squared-length share c1​ = 0.447 20% c2​ = 0.894 80% 1 (λ1​) 2 3 (λ2​) Unit-input score: 2.6 The score for every direction The dot is the current input λ1​ = 1 λ2​ = 3 −180° −90° 0° 90° 180° 0 1 2 3
18.4°

Score for this unit input: 2.6

20% of squared length along u₁; 80% along u₂.

Drag to turn the arrow around the circle; the angle is measured from the positive horizontal axis. Watch the dashed remainder vanish at the eigenvectors, the pieces trade shares, and the dot ride the curve.

Every direction gets the same input length. Ordinary coordinates, as in Chapter 9. Left: apply M to the unit input and drop the output's shadow back onto it. Right: split the same input into its eigenvector pieces; their shares and the score update as the arrow turns. Bottom: the same score for every direction at once, with a dot marking the current input.
What if we resize a copy of the input?

Keep the circle experiment at length one. Here we resize a separate copy of its current input. The copy and all its pieces change length together; the unit reference arrows stay fixed.

1
Copy's squared length1
Copy's raw score2.6
Copy's quotient2.6

Changing the copy's length changes its raw score and its squared length by the same factor, so their ratio doesn't budge. The circle picture above keeps showing the unit input.

For the original x, halving its length lowers the score from 26 to 6.5 and the squared length from 10 to 2.5. Both ratios are still 2.6. The shrink ray strikes out.