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University · Linear algebra

Find directions a matrix preserves

Separate a vector into two perpendicular eigen-directions and watch each scale independently.

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

Watch the relationship

Paused
Find directions a matrix preserves. Current vector image: (1, 0). Current eigen-scales: 1, 1. Final determinant: 3Two directions keep their lines(1, 1)(1, −1)Amber: input image · Negative scale reverses
All vectors are real and the two displayed eigen-directions are orthonormal after division by √2. Playback interpolates from I to A; it is not a dynamical-system simulation. When λ₋ = 0, A is singular. The zero vector is never an eigenvector.

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.

Current vector image
(1, 0)
Current eigen-scales
1, 1
Final determinant
3

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From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

An eigenvector is a nonzero vector whose image lies on the same line through the origin. For this symmetric matrix, the diagonal directions (1, 1) and (1, −1) form an eigenbasis. Any vector can be built from those two directions, then transformed by scaling each part.

A relationship to keep

A = ½[[λ₊ + λ₋, λ₊ − λ₋], [λ₊ − λ₋, λ₊ + λ₋]]

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

  1. STEP 1

    Choose an input direction

    The amber input vector has length one. The navy and teal diagonal lines are eigen-directions, so vectors on those lines will remain on the same lines after the map.

  2. STEP 2

    Scale the two modes

    Playback gradually applies the selected transformation. The colored arrows show the changing images of unit eigenvectors. A negative eigenvalue reverses the corresponding arrow.

  3. STEP 3

    Compare input and output

    The moving amber arrow is the image of the input. Unless the input lies in an eigen-direction, or the matrix scales every direction equally, its direction can change.

Your turn to explain

Make a prediction. Test your reasoning.

Set λ₊ = 3 and λ₋ = 1. What is A(1, 0)?

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

Compare your explanation

(2, 1). Here A = [[2, 1], [1, 2]], so its first column is the image of (1, 0).

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Algebra

Find directions a coupled system preserves

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Compute a large matrix power through its eigenbasis

Use diagonalization to avoid repeated matrix multiplication.