B Why the hate on determinants?

Buffu

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 ?

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fresh_42

Mentor
2018 Award
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.
Hard to tell without the context. And very likely not really a good idea to say something like this.
Is determinants of no use in Maths ? which tool has taken its place in algebra ? And why this happened ?
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.

scottdave

Homework Helper
I have learned something new. I did not know that books were teaching the uselessness of determinants. I don't know why they would say that.
Are they saying that it is useless to learn the steps involved in calculating the determinant, or that the determinant has no use. One use of a determinant, is to tell if a set of equations is independent, or not. (check to see if determinant equals zero).

If the determinant is not equal to zero, then you can use Cramer's method to solve the system. This involves taking more than one determinant. As Excel and other spreadsheets have a MDTERM() function to calculate determinants, you can reduce the process of solving a system to keying some coefficients into a spreadsheet.
Wtih larger matrices, I would not recommend trying to tackle a determinant by hand, you should know how to go about doing it.

Here is how Cramer's method works http://www.purplemath.com/modules/cramers.htm

Mark44

Mentor
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.
Another example use is in finding the eigenvalues of a matrix.

Again, the context of why Strang et al. said that would be helpful.

Homework Helper

FactChecker

Gold Member
2018 Award
Determinants have a good interpretation as the effect of the matrix on an oriented n-dimensional volume. Saying that determinants are useless is like saying that areas and volumes are useless. I wouldn't give it too much thought. A person would be making a great mistake to omit learning about determinants.

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mathwonk

Homework Helper
i advise you to ignore these comments and learn determinants as well as possible. the most fundamental invariant of any linear operator is its characteristic polynomial, and the determinant is the constant term of this polynomial. nobody can change this fact of nature. even those who rail against determinants know them very well, and are just (in my opinion) showing off how clever they are at thinking of an alternative approach to some aspects of the theory. Determinants measure volume; who would argue that volume is not important? take a look at the change of variables formula for integrals in several variables. the only troublesome aspect of determinants is that they are hard to compute, and hard to define. so what? they are a basic fact of life in mathematics. they underlie the super important tool of differential forms and "wedge product". they define the beautiful plucker embedding of the space of lines in projective 3 space to the quadric hypersurface in P^5. there may be clever ways to get around them in certain settings, but anyone who does not know them pretty well is ignorant of something essential.

i see now that fact checker said it better and shorter.

Skins

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 ?
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.

Erland

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
Axler wrote a paper about banning deteminants, available on the net:
http://www.axler.net/DwD.pdf
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.

willem2

. 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.
But no one uses determinants to compute eigenvalues for any matrix of size >=5. Look up the QR method.

atyy

Volume is useless now? http://mathinsight.org/relationship_determinants_area_volume

https://en.wikipedia.org/wiki/Slater_determinant

https://arxiv.org/abs/0812.2691 (Pfaffian is a determinant)

https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
On teaching mathematics by V.I. Arnold
"The determinant of a matrix is an (oriented) volume of the parallelepiped whose edges are its columns. If the students are told this secret (which is carefully hidden in the purified algebraic education), then the whole theory of determinants becomes a clear chapter of the theory of poly-linear forms. If determinants are defined otherwise, then any sensible person will forever hate all the determinants, Jacobians and the implicit function theorem."

Stephen Tashi

The debate on determinants is not whether they should be taught, but when in the sequence of instruction they should be taught.

For example, if students are drilled in solving systems of equations by determinants, the can develop a mental resistance to learning about the concepts of linear independence, Gaussian elimination, elementary matrices etc.

It is important to use determinants to define theoretical concepts like characteristic polynomials. But it would be misconception to think that numerical computations involving large matrices implement the theoretical concepts by computing the value of a determinant exactly as the definition of a determinant specifies. So numerical analysis texts emphasize that the definition of a determinant is not the practical way to compute determinants of large matrices -and that using determinants is not usually the practical way to solve a system involving a large number of linear equations. I wouldn't call those practical statements a "hate" for determinants.

StoneTemplePython

Gold Member
This is not a new thing, by the way.

In the 1960's a prof at U Texas did an hour long video called "Who Killed Determinants?". I couldn't find a free copy of the video on Youtube, though, so I may never know.

http://search.library.utoronto.ca/details?214517

Skins

Axler seems to imply that there is definitely a need and a time and place for determinants. However, certain proofs, ideas, and computations can be accomplished in more elegant ways without determinants than with. I tend to agree with this philosophy. However, determinants are by no means useless nor are they obsolete. Indeed they do have a well deserved place in this most fascinating thing we call Mathematics.

Math_QED

Homework Helper
From a theoretic point of view, determinants are very useful, since it translates all information of a matrix to one scalar, from which one can say that an inverse exists, a system of equations is independent, etc. The determinant not only occurs in linear algebra, but also in other fields, like analysis (calculus). An example that comes to mind is the Jacobian determinant, which is used as scaling factor in coordinate transformations when calculating complicated integrals in more than 1 variable. Another analysis determinant is the Wronskian, which is for example used in the theory of differential equations (to see whether a system of differential equations has a unique solution).

The determinant is also a useful tool to memorise how to compute the cross product of 2 vectors, or the curl of a vector field (as an application of the cross product).

I would not bother when an author says something like this. Just ignore it and move on.

mathwonk

Homework Helper
here is another example of where the concept of a determinant is absolutely fundamental. This is the context of wedge products. Briefly, there is a construction that changes an n dimensional space into a one dimensional space, whose elements are roughly parallelepipeds with n sides, two such being equivalent if they have the same oriented volume. Then a linear map between two n dimensional spaces induces a linear map between the corresponding two one dimensional spaces, which must be multiplication by a scalar. that scalar is the determinant of the original transformation, and measures the change in volume induced by the original transformation. By analogy one may call the induced one dimensional space the determinant, or more preciely the nth wedge product, of the original n dimensional space. In geometry the most basic construction on a smooth n-manifold is its tangent bundle, a family of n dimensional spaces. The induced determinant bundle, the nth wedge product of the dual of the tangent bundle, or the bundle of n-forms, is a family of one dimensional spaces, called the canonical line bundle of the manifold, and is the most fundamental invariant of the manifold. The zero locus of any section of this bundle is called a canonical class of the manifold. To illustrate its importance, the degree of this class on a surface of genus g is 2g-2, so it determines the genus. So the idea of a determinant is so basic that it persists throughout geometry, and I believe the best place to begin to learn about it is at the elementary level. I.e. if you know about elementary determinants you have some chance of grasping their deeper generalizations.

Here is a wikipedia article on the canonical bundle, using the term "determinant" in this way:

https://en.wikipedia.org/wiki/Canonical_bundle

here is a nice elementary book on geometry of forms by David Bachman.

https://www.amazon.com/dp/0817683038/?tag=pfamazon01-20

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mathwonk

Homework Helper
As an example of the usefulness of the canonical bundle in making calculations, one can compute the genus of a projective algebraic plane curve, say over the complex numbers. Then the genus is actually the topological genus of the real surface underlying the complex curve. In general the (degree of the) canonical class on a curve of genus g equals 2g-2, i.e. the negative of the euler characteristic.

By computing an actual 2 form one sees that the canonical class of the plane itself is O(-3), i.e. a standard 2 form has a triple pole along a line, and no zeroes. Then there is a wonderful formula for how canonical divisors restrict, the adjunction formula. This says that for a curve of degree d in the plane, the canonical class on the curve equals O(d-3).

Hence for a line we get -2, agreeing with the fact that the 1-form dz has no zeroes in the affine plane and one double pole at infinity, under the substitution z = 1/w. For a plane conic we get O(-1) but this means the divisor is cut out on a conic by intersecting with a curve of degree 1, i.e. a line. Since a line meets a conic twice we again get O(-2) on the conic itself, which agrees with the fact that a conic is isomorphic via projection with a line, and both are homeomorphic to the sphere, with genus zero. On a plane cubic we get O(0), or the trivial class, agreeing with the fact that a smooth cubic is homeomorphic to a torus of genus one, and euler characteristic zero. As exercise check that a (smooth) plane curve of degree4 has genus 3.

We can also compute the genus of a curve cut by two surfaces in P^3, complex projective 3 space, using the higher adjunction formula, that the curve cut by surfaces of degree d and e, has canonical class of degree equal to: de(d+e-4). Thus two quadratic surfaces cut a curve of degree 4 and class zero, hence again a torus. Indeed projecting such a curve to the plane from one point of itself lowers the degree by one and gives an isomorphism with a plane cubic.

Intersecting two surfaces of degrees 2 and 3 gives (exercise) a curve of genus 4.

If we pass two cubic surfaces through a quartic space curve C of genus g = one, the full intersection has degree 9 hence consists of that quartic plus some other curve C’ of degree 5. We can even compute the genus of that residual curve C’ by the formula (obtained by subtracting two adjunction formulas):

2(g’-g) = (5-4)(3+3-4) = 2, so the other curve C’ has genus g’ = g+1 = 2, and is hence a space quintic of genus 2. Note that projecting this curve to the plane from a point of itself gives a plane quartic which should have genus 3. Since it has only genus 2, the plane projection must cross itself once, lowering the genus. This means that every point of the space quintic, that we might choose to project from, must lie on a trisecant, something not obvious to me at the moment. It also suggests that the canonical class on C' of degree 2g'-2 = 2, is swept out residually on the space quintic by the pencil of planes through such a trisecant. I.e. each such plane cuts the quintic in 3 fixed points on the trisecant, and further in two moving points. I don't see why this follows from adjunction either at the moment, but it is true for general reasons.

These calculations are all manifestations of the concept of differential n - form, i.e. the "determinant" of the space of one forms dual to tangent vectors. These computations were known to European geometers close to 150 years ago, but seem simpler today through the systematic use of the canonical bundle of n - forms. In particular I found this last "residual genus" formula mysteriously presented in a wonderful old book by Semple and Roth on classical algebraic geometry, and was able to derive it myself (just yesterday) this way.

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Krushnaraj Pandya

Gold Member
They may be very useful. But any high-school level textbook in India has the most tedious calculations in this chapter including finding inverse matrices by hand, so the 'hatred' for it

scottdave

Homework Helper
I took the linear algebra MOOC course (LAFF - Linear Algebra, Foundations to Frontiers through edX and University of Texas) in Fall of 2018.
I believe they talked briefly about determinants. I believe one use of them is in computing eigenvalues. Another is Cramers method in solving a linear system, and as an extension of that, to determine if all of the equations in a linear system are independent.

The determinants can be quite costly (computation wise), especially for large systems. So they showed other methods to achieve the desired results.

stevendaryl

Staff Emeritus
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 ?
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?

WWGD

Gold Member
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?
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?

Cryo

Gold Member
I like Axler's book very much. I like the emphasis on vector space and linear independence. Never actually reached the chapter on determinants in his book :-)

Here is how you define an eigenvalue without determinants. Given an operator (matrix) $\mathbf{T}$ the eigenvalue $\lambda$ is such scalar that the following kernel (null-space) is not empty:

$Ker\left(\mathbf{T}-\lambda \mathbf{Id}\right)\neq\emptyset$

where $\mathbf{Id}$ is identity. The eigenvectors are those vectors that span this non-empty kernel. The nice thing about this definition is that it is easy to extend to generalized eigenvectors.

I think Axler's point was that determinants are like black-boxes. It is hard to understand why they are so useful. Also, as some other people already commented. Computing determinants for large matricies may be not a good way to solve common problems in linear algebra.

mathwonk

Homework Helper
Do people hate on addition, given that the task of adding up a million numbers is hard by hand? 2x2 determinants are not so hard, just as adding two digit numbers is fairly easy. some of us even think basic subtraction is hard (have you balanced your checkbook lately?), but we don't dismiss it. the point is, as a scientist, you may not ignore any aspect of nature, whether or not it seems easy to deal with.

Ray Vickson

Homework Helper
Dearly Missed
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.
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.

"Why the hate on determinants?"

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