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Undergraduate · Diagonal eigenmodes

Diagonal eigenmodes: Uniform doubling

Diagonal eigenmodes: investigate uniform doubling with first eigenvalue a = 2; second eigenvalue b = 2.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Diagonal eigenmodes: Uniform doubling. Image x coordinate: 1. Image y coordinate: 1. Current determinant: 1. Final determinant: 4Two independently scaled directionsxy0Dashed: unit squareFilled: current imageEqual axis scales · coordinates in the readouts
The animation shows interpolation of matrices, not matrix powers or a physical trajectory. The input is fixed at (1,1). Determinants can be zero or negative, and signed area differs from ordinary nonnegative area.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Image x coordinate
1
Image y coordinate
1
Current determinant
1
Final determinant
4

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

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. This investigation starts with First eigenvalue a = 2; Second eigenvalue b = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

A=diag(a,b); B(t)=(1−t)I+tA

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Start with the input (1,1) and the identity map. The starting case is “Uniform doubling.”

  2. STEP 2

    Follow the changing quantity

    Interpolate from the identity to the selected diagonal matrix. Track the current image and the signed product of its two scale factors.

  3. STEP 3

    Explain and test the result

    At the endpoint, compare determinant sign, singularity, and the two eigenvalues. An eigenvalue is a scale, not a coordinate of every input vector.

Your turn to explain

Make a prediction. Test your reasoning.

Keep First eigenvalue a = 2; Second eigenvalue b = 2. Pause the timeline at 20%. Given image x coordinate = 1.2, calculate image y coordinate, current determinant, final determinant. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Current diagonal scales are 1+0.2(2−1)=1.2 and 1+0.2(2−1)=1.2. Their product is 1.44; the final matrix determinant is (2)(2)=4. Results: Image y coordinate: 1.2; Current determinant: 1.44; Final determinant: 4. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Algebra

Find directions a coupled system preserves

Compute eigenvalues and eigenvectors of a symmetric coupling matrix.