Undergraduate · Diagonal eigenmodes · 94 of 650
Diagonal eigenmodes · First eigenvalue a=-1; Second eigenvalue b=-0.5
Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=-1; Second eigenvalue b=-0.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A diagonal linear map is A=diag(-1,-0.5). Apply it to (1,1) and the unit square. Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image.
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A starting point
A diagonal map scales each coordinate separately. Multiply the two scale factors for the signed determinant, then take its absolute value for area.
Work through the reasoning
Step 1
Identify the model and target
A diagonal linear map is A=diag(-1,-0.5). Apply it to (1,1) and the unit square. Use the completed matrix transformation at the animation endpoint. The governing relation is A=diag(a,b); B(t)=(1−t)I+tA. A diagonal map scales each coordinate separately. Multiply the two scale factors for the signed determinant, then take its absolute value for area.
Step 2
Substitute and calculate
Current diagonal scales are 1+1(-1−1)=-1 and 1+1(-0.5−1)=-0.5. Their product is 0.5; the final matrix determinant is (-1)(-0.5)=0.5.
Step 3
Check the mathematical meaning
The determinant is (-1)(-0.5)=0.5, while unsigned area is 0.5. The positive determinant preserves orientation even if both coordinate directions reverse. Image x coordinate: -1; Image y coordinate: -0.5; Current determinant: 0.5; Final determinant: 0.5. Decimal values are rounded, so use unrounded intermediate values.
The answer
Current diagonal scales are 1+1(-1−1)=-1 and 1+1(-0.5−1)=-0.5. Their product is 0.5; the final matrix determinant is (-1)(-0.5)=0.5. The determinant is (-1)(-0.5)=0.5, while unsigned area is 0.5. The positive determinant preserves orientation even if both coordinate directions reverse. Animation check: Image x coordinate: -1; Image y coordinate: -0.5; Current determinant: 0.5; Final determinant: 0.5. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
A diagonal map scales the coordinate axes independently, so those axes are eigen-directions. Applying the map to (1,1) makes the two scale factors visible together. A negative factor reverses orientation along an axis, while a zero factor loses a dimension. A diagonal linear map is A=diag(-1,-0.5). Apply it to (1,1) and the unit square. Use the completed matrix transformation at the animation endpoint. The requested state occurs at 100% playback; use the exact target stated in the question for your calculation.
A=diag(a,b); B(t)=(1−t)I+tA
03 · Reflect and transfer
Explain what changes and why.
Compare a negative determinant with a zero determinant. Which represents a reflection and which represents loss of dimension?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.