Symmetric matrices: perpendicular eigenvectors
On slides 2 to 4 the eigenvectors could point any way: 11 and 1−2 meet at a slant. Symmetric matrices (A = Aᵀ, from lesson 3) are much tidier. Think of pizza dough pulled harder east–west than north–south. The two pull directions don't turn, and they are at a right angle. Every symmetric matrix acts like that: it stretches or flips along perpendicular axes, and does nothing else. Its eigenvalues are real numbers, and its eigenvectors can be chosen perpendicular and of length 1. Put those unit eigenvectors in the columns of Q. Then Q is an orthogonal matrix (lesson 5: a square matrix with orthonormal columns, so QᵀQ = I), its inverse is simply Qᵀ, and A = QΛQᵀ. This fact is called the spectral theorem.
Lesson 7 · slide 5Symmetric matrices- Every eigenvalue of a symmetric matrix is a real number (never complex, unlike the rotation on slide 1).
- Eigenvectors for different eigenvalues are perpendicular.
- Scaled to length 1, they form the columns of an orthogonal Q (lesson 5): Q⁻¹ = Qᵀ.
- So A = QΛQᵀ. (This result is called the spectral theorem.) Every symmetric matrix is a stretch along perpendicular axes.
Stretching along perpendicular axes
Take the symmetric A = 3113: row 1 and column 1 hold the same numbers. Trace 6, det 9 − 1 = 8:
A circle shows the same thing. The unit circle becomes an ellipse whose long and short axes lie exactly on the eigenvector lines. The long half-axis is 4, the short one is 2.
Why the eigenvectors are perpendicular
Take Ax = λ₁x and Ay = λ₂y with λ₁ ≠ λ₂. Compute xᵀAy two ways (the transpose rules are from lesson 3):
Without symmetry the second line fails, and so does the right angle. The matrix 4123 from slide 2 is not symmetric, and its eigenvectors give 11 · 1−2 = −1 ≠ 0.
Unit length: from X to Q
Divide each eigenvector by its length, √2 here, to get unit vectors. They are perpendicular and of length 1, so the matrix Q with them as columns has QᵀQ = I (lesson 5):
Length 1 matters. Without the √2, QᵀQ = 2I and QΛQᵀ comes out as 2A. That is the exercise below.
A sum of simple pieces
First multiply QΛQᵀ out directly, row times column as in lesson 2. QΛ multiplies column 1 of Q by 4 and column 2 by 2:
The same product can be grouped another way: column 1 of QΛ times row 1 of Qᵀ, plus column 2 of QΛ times row 2 of Qᵀ (lesson 8, slide 3 uses this a lot). That gives A = λ₁q₁q₁ᵀ + λ₂q₂q₂ᵀ. A column times a row is a full matrix: entry (i, j) is entry i of q times entry j of q.
Each qqᵀ is the projection matrix onto the line through q (lesson 5). So A splits x onto the two lines, stretches each piece by its λ, and adds them back:
Check an eigenvector by multiplying
Keep reading with the full course
Finish lesson 7 and you will be able to: tell whether a system settles down or blows up.
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Examples
Example 1 · A diagonal matrix
A = 5001: Q = I, Λ = A. The axes are already perpendicular eigenvectors.
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