Hard to tell without the context. And very likely not really a good idea to say something like this.Whty does most books on linear algebra have something like "Determinants are useless now".I have seen this in Strang, Friedberg and Axler's book.
Au contraire! The opposite is true. Determinants are very important in various fields: solving linear equations, shortest way to say and test a matrix is regular, field theory, geometry, theory of algebraic groups, multivariate calculus, and more if I would search for it.Is determinants of no use in Maths ? which tool has taken its place in algebra ? And why this happened ?
Another example use is in finding the eigenvalues of a matrix.Au contraire! The opposite is true. Determinants are very important in various fields: solving linear equations, shortest way to say and test a matrix is regular, field theory, geometry, theory of algebraic groups, multivariate calculus, and more if I would search for it.
I assume you mean Dr. Gilbert Strang ? Where did you see this in Dr. Strang's book ? I can't speak for Friedberg or Axler but, I have reviewed Dr. Strang's lectures and have also read his book which covers determinants in depth and nowhere does he imply that determinants are useless and/or are no longer needed. Can you cite a particular lecture or a writing from his book where he says that determinants are useless ? I see nothing of the kind.Why do most books on linear algebra have something like "Determinants are useless now".I have seen this in Strang, Friedberg and Axler's book.
Are determinants of no use in Maths ? which tool has taken its place in algebra ? And why this happened ?
Axler wrote a paper about banning deteminants, available on the net:I have not read this book, so I was curious about it. I did some searching and came across some opinions on Quora that you may find interesting. https://www.quora.com/What-do-mathematicians-think-of-Axlers-Linear-Algebra-Done-Right
But no one uses determinants to compute eigenvalues for any matrix of size >=5. Look up the QR method.. He doesn't say a word about how to calculate eigenvalues without determinants. It is not at all clear how to do this, and I doubt that it is even possible to do in any feasible way.
https://www.math.brown.edu/~treil/papers/LADW/LADW.htmlI have not read this book, so I was curious about it. I did some searching and came across some opinions on Quora that you may find interesting. https://www.quora.com/What-do-mathematicians-think-of-Axlers-Linear-Algebra-Done-Right
I don't know what that quote means, but maybe they're talking about computer algorithms that accomplish many of the goals of matrix analysis (computing eigenvalues, inverses, etc.) without first computing the determinant?Why do most books on linear algebra have something like "Determinants are useless now".I have seen this in Strang, Friedberg and Axler's book.
Are determinants of no use in Maths ? which tool has taken its place in algebra ? And why this happened ?
Maybe it has to see with the fact that running time grows fast with n for finding determinants, so other techniques are used for large matrices?I don't know what that quote means, but maybe they're talking about computer algorithms that accomplish many of the goals of matrix analysis (computing eigenvalues, inverses, etc.) without first computing the determinant?
I read some entries by Prof. Robert Israel some years ago in a computer-algebra group. He said that in practice (at least for totally numerical examples) eigenvalues are typically not found by finding roots of the characteristic polynomial, but rather, by other methods. He went on to say that some posted methods for finding polynomial roots proceed by converting the polynomial into the characteristic equation of some matrix, then using other eigenvalue algorithms to find the roots.Axler wrote a paper about banning deteminants, available on the net:
It's interesting and he has some good points. But the main weakness is that he just proves theorems. He doesn't say a word about how to calculate eigenvalues without determinants. It is not at all clear how to do this, and I doubt that it is even possible to do in any feasible way.