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A little observation: You repeatedly use language that fails to distinguish definitions & properties vs. effective computational methods. For example, in secti

A little observation: You repeatedly use language that fails to distinguish definitions & properties vs. effective computational methods. For example, in section G: > To find the eigenvalue of a matrix we start from the eigenvalue equation .... Solving the resulting equation is one way of computing eigenvalues. But it might not be the one you want to use in some practical situation. Just before that, in section F: > The determinant of a matrix, ... serves to check if a matrix is invertible or not. It is true that a square matrix is invertible iff is has nonzero determinant. It certainly is not true that, for a matrix of any size, computing the determinant is a good method for checking whether a matrix is invertible.

shitlord 13y ago
> It certainly is not true that, for a matrix of any size, computing the determinant is a good method for checking whether a matrix is invertible. Can you explain this part? I don't understand what you mean, and I have an Advanced Linear Algebra final in a few days, so I should probably know this stuff. I thought the determinant was not defined for non-square matrices. Also, I thought that inverses only existed for square matrices (implying that the determinant check is a good way of testing invertability).
- Steuard 13y ago
  It's a valid way to test whether a (square) matrix has an inverse, but computing big determinants is a lot of effort (both from a "student work" perspective and from a "computational cost" perspective). The point is that in many cases, there are enormously more efficient ways to find an inverse (or, more generally, to solve matrix problems with inverses in them).
  - shitlord 13y ago
    Oh, okay. Thanks for the clarification.
Steuard 13y ago
> Solving the resulting equation is one way of computing eigenvalues. But it might not be the one you want to use in some practical situation. Related fun fact: Solving the eigenvalue equation is (in general) impossible in closed form for matrices 5x5 or larger, because there's no closed form solution to the quintic. So some other algorithm for finding eigenvectors and eigenvalues is often absolutely required in practice.
- ggchappell 13y ago
  Well, that depends on whether you want the exact eigenvalues.

shitlord 13y ago

> It certainly is not true that, for a matrix of any size, computing the determinant is a good method for checking whether a matrix is invertible. Can you explain this part? I don't understand what you mean, and I have an Advanced Linear Algebra final in a few days, so I should probably know this stuff. I thought the determinant was not defined for non-square matrices. Also, I thought that inverses only existed for square matrices (implying that the determinant check is a good way of testing invertability).

Steuard 13y ago
It's a valid way to test whether a (square) matrix has an inverse, but computing big determinants is a lot of effort (both from a "student work" perspective and from a "computational cost" perspective). The point is that in many cases, there are enormously more efficient ways to find an inverse (or, more generally, to solve matrix problems with inverses in them).
- shitlord 13y ago
  Oh, okay. Thanks for the clarification.

Steuard 13y ago

> Solving the resulting equation is one way of computing eigenvalues. But it might not be the one you want to use in some practical situation. Related fun fact: Solving the eigenvalue equation is (in general) impossible in closed form for matrices 5x5 or larger, because there's no closed form solution to the quintic. So some other algorithm for finding eigenvectors and eigenvalues is often absolutely required in practice.

ggchappell 13y ago
Well, that depends on whether you want the exact eigenvalues.