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Eigenvalues and eigenvectors

Symmetric matrices

A real symmetric matrix has real eigenvalues and eigenvectors that can be chosen perpendicular, so it stretches or flips the plane along perpendicular axes. This topic reads positive definite matrices as bowls and ellipses, then extends the ideas to complex vectors and the Fourier matrix.

Lessons on this topic

Animation: the unit circle warped by S = [5 2; 2 2] into an ellipse. Its perpendicular axes: 6 along 26.57 degrees and 1 along 116.57 degrees.Animation: the unit circle warped by S = [5 2; 2 2] into an ellipse. Its perpendicular axes: 6 along 26.57 degrees and 1 along 116.57 degrees.

Intermediate · 25 minYou are hereRead

Symmetric matrices

Watch a symmetric matrix squeeze a circle into an ellipse whose axes always meet at a right angle, and write it as stretches along those axes.

Animation: the surface z = 2x² + 2sxy + 2y² in a slowly turning 3-D view, at s = 3. It is a saddle. Along x = y it curves with eigenvalue 5, along x = −y with eigenvalue −1.Animation: the surface z = 2x² + 2sxy + 2y² in a slowly turning 3-D view, at s = 3. It is a saddle. Along x = y it curves with eigenvalue 5, along x = −y with eigenvalue −1.

Intermediate · 25 minYou are hereRead

Positive definite matrices

Draw xᵀAx as a surface and tell a bowl from a valley or a saddle, with four tests that ask the same question, and factor a bowl by Cholesky.

Columns 1 and 1 of the 8 by 8 Fourier matrix as rings of arrows, and the chain of their 8 products laid so far. The sum is 8.Columns 1 and 1 of the 8 by 8 Fourier matrix as rings of arrows, and the chain of their 8 products laid so far. The sum is 8.

Intermediate · 25 minYou are hereRead

Complex vectors and the Fourier matrix

Measure complex vectors with the conjugate, meet Hermitian and unitary matrices, and see why the Fourier matrix’s columns are perpendicular and diagonalise every circulant.

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