Estimate parameter [color=#1e84cc][i]a[/i][/color] so that the matrix [i]B[/i] represents revolution about origin [i]O[/i] = (0,0). Find all fixed points and directions.[br][center][img]data:image/png;base64,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[/img][/center]
1. method: All rotation about origin have matrix form R(Φ), where Φ is rotation angle. [br][br][img]data:image/png;base64,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[/img][br]Comparing elements of matrices R(Φ) and B yields [math]sin\text{Φ}=\frac{\sqrt{2}}{2}\Rightarrow a=cos\text{Φ}=\frac{\sqrt{2}}{2}[/math].[br]2. method: Rotation is a direct isometry, hence |B|=1, i.e. [math]a^2+\frac{2}{4}=1[/math].[br]3. method (experimental): Use tool slider[icon]/images/ggb/toolbar/mode_slider.png[/icon]for unknown parameter [color=#1e84cc][i]a[/i][/color]. Define one parameter family of matrices B([color=#1e84cc][i]a[/i][/color]). [br][code]B={{a,-sqrt(2)/2},{sqrt(2)/2,a}}[/code][br]Draw arbitrary object [color=#1e84cc][i]A[/i][/color] (point, segment or picture) and its image [color=#1e84cc][i]A[/i][/color][color=#1e84cc]'[/color] - GeoGebra command [code]ApplyMatrix(B,A)[/code]. Observe the effect of changing the slider [color=#1e84cc][i]a[/i][/color] and estimate correct value for parameter [color=#1e84cc][i]a[/i][/color]. [br][br]Experimental method is efficient for determination of fixed point and directions. Compare the position of arbitrary movable point [i][color=#1e84cc]A[/color][/i] and its image [i][color=#1e84cc]A[/color][/i]. Find out the location where points coincide, [i][color=#1e84cc] A[/color][/i] = [color=#1e84cc][i]A[/i]'[/color]. There is the fixed point of transformation. The same method applyed on line [color=#1e84cc][i]f[/i][/color] gives you fixed direction. You should find the position where [i][color=#1e84cc]f[/color][/i] is parallel with image [color=#1e84cc][i]f'[/i][/color].[br][br][color=#444444][size=150]Fixed points x'=x[/size][/color][center][img]data:image/png;base64,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[/img][/center]where E is identity matrix. First determine substraction [code]BE=B-Identity(2)[/code] and than use GeoGebra tool [code]ReducedRowEchelonForm(BE).[/code] This eliminates non diagonal elements by row operations (= Gaussian elimination).[center][img]data:image/png;base64,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[/img][/center]Using back-substitution, unknowns [i]x, y[/i] can be solved for. Solution x = 0 and y = 0 gives only one fixed point FP = (0,0). [br][br]
Eigenvector [i]x[/i] of matrix [i]B[/i] has invariant direction in transformation defined by matrix [i]B.[br][br][center][img]data:image/png;base64,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[/img][/center][/i][size=100]Matrix (B-λE) must be singular for non trivial solutions x, but Det(B-λE)=0 has no real solutin. Rotation has no fixed direction.[/size]