michael_nielsen

The determinant of a matrix is how much the matrix expands or contracts space. More precisely: it's the volume of the parallelepiped spanned by the columns of the matrix. (Ditto the rows.) This, btw, is why det 0 => non-invertible: one or more spatial dimensions has collapsedhttps://twitter.com/ZachWeiner/status/1073671743846395905 …

That's one way. There are many other useful equivalent ways of thinking of the det: (1) As the product of the eigenvalues (for normal matrices); (2) as the product of the singular values (up to sign) for any matrix; (3) as the value of the wedge product of the columns c1^c2^ ...

It's the product of the eigenvalues for any matrix. Not just normal ones.

The 2 by 2 matrix [ [2, 0], [1, 2] ] has determinant 4, and just a single eigenvalue, 2.

Of course, if you want to raise the eigenvalues to the power of their algebraic multiplicities, then you do get the determinant. That's another nice way of thinking about it. But it wasn't what my tweet was referring too (geometric multiplicities).

Well we don't disagree then except on semantics. I think of n x n matrix always having n eigenvalues and Det is their product. I think that's very standard. But you're using a different convention.

The matrix I described above is 2 by 2, but has just a single eigenvector. So you're wrong about n x n matrices having n eigenvalues.