Gram Matrices: Two of Them, One Set of Eigenvalues
An interactive live film: A^T A and A A^T share their nonzero eigenvalues, the squared semi-axes of the ellipse A sends the unit circle to. Drag a column until the ellipse is a circle.
Narration
A matrix A with three rows and two columns: its columns are two vectors in space, a zero and a one.
It has two Gram matrices. A transposed A is two by two: the inner products of the columns. A A transposed is three by three: the inner products of the rows.
The rest of the narration is in the book's Companion.
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