EIGENVALUES & EIGENVECTORS: THE DIRECTIONS A MATRIX DOESN’T ROTATE

In Lesson 3's Worked Example 3, we discovered something curious: applying the "swap" matrix to the vector left it completely unchanged. That wasn't a coincidence, and it wasn't a special property of that one matrix — it's an instance of a general and extremely important phenomenon: every matrix has certain special directions that it doesn't rotate, only stretches or shrinks (or leaves entirely alone). These special directions are called eigenvectors, and the amount of stretching/shrinking along them is called the eigenvalue.

This concept becomes essential starting in Module 4 (certain quantum states are the "natural resting states" of certain gates) and becomes central to Module 10, Lesson 5 ("Phase Kickback"), one of the most important tricks in the entire field.

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Building Intuition First: The Rubber Sheet Analogy

Imagine stretching a rectangular rubber sheet along one direction — say, pulling it wider left-to-right, while leaving the top-to-bottom direction alone. Most arrows drawn on the sheet, if stretched along with it, will end up rotated to point in a somewhat different direction than before. But two special directions won't rotate at all: an arrow pointing purely left-right will just get longer, still pointing purely left-right; an arrow pointing purely up-down will stay exactly the same length and direction, since that direction wasn't stretched at all.

Those two special, un-rotated directions are the eigenvectors of this stretching transformation. The amount each one got scaled by is the corresponding eigenvalue.

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The Formal Definition

Notice the pattern: normal matrix-vector multiplication () can produce a wildly different-looking vector. But for an eigenvector specifically, the output is just a scaled copy of the input — the simplest possible thing a matrix could do to a vector.

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Worked Example 1: Confirming an Eigenvector We Already Found

We already discovered, informally, that is an eigenvector of in Lesson 3. Let's confirm using the formal definition:

The output equals exactly times the input — so this is an eigenvector with eigenvalue .

Worked Example 2: Finding a Second Eigenvector of the Same Matrix

Let's test (which, from Lesson 5, is |−⟩) against the same matrix :

Is this a scaled copy of the input? The input was , and the output is , which is exactly times the input:

Yes! This is also an eigenvector, with eigenvalue .

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Finding Eigenvectors and Eigenvalues Systematically

The examples above worked because we already suspected the answer. Here's the general, systematic method.

Step 1: The Characteristic Equation

Starting from , we can rewrite this as . For a nonzero vector to satisfy this, the matrix must not be invertible — and a 2×2 matrix fails to be invertible exactly when its determinant is zero.

For a 2×2 matrix , the determinant is:

So we need to solve .

Worked Example 3: Finding Eigenvalues of from Scratch

Step 1 — form :

Step 2 — compute the determinant and set it to zero:

Expanding:

Setting this equal to zero:

Step 3 — find the eigenvector for . We need :

This gives , so , and can be anything nonzero — simplest choice . Eigenvector: .

Step 4 — find the eigenvector for . We need :

This gives , so , and can be anything nonzero — simplest choice . Eigenvector: .

Conclusion: The matrix (which you verified was unitary in Lesson 4, and which Module 4, Lesson 3 will formally call the Pauli-Z gate) has |0⟩ as an eigenvector with eigenvalue , and |1⟩ as an eigenvector with eigenvalue . This confirms, from pure matrix algebra, that |0⟩ and |1⟩ are the "natural resting states" of this particular matrix — exactly matching the fact that they sit on the Z-axis of the Bloch sphere, the same axis this matrix is named after.

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