# Physical applications of matrices and determinants

Other than notation devices, I don't see any direct application of matrices/determinants in physics. For example, they are just a different way to write a partial derivative and determinants find if they can be explicitly solved if written down as simultaneous equations. Calculus, for instance, can be directly applied to physical problems, but I don't know of any other application of matrices other than representing equations in a different notation. And in most of the cases like vector products, you just realise that a huge term can just be written down as a determinant, so it is essentially a notational tool. They are used in tensor calculus, but for similar reasons.

Can someone please guide me on more applications with good sources?

• Comment to the question (v2): This seems like a list question: Linear algebra, matrices and determinants are used in virtually all areas of physics. – Qmechanic Aug 18 '14 at 16:17
• It seems a bit silly to say that matrices are "just notational tools" even in the examples you gave. A lot of the ways you actually use matrices to solve the problems mentioned would look totally bizarre without the linear transform point of view. Have you studied eigenvectors and eigenvalues? – Robert Mastragostino Aug 18 '14 at 16:29
• I suppose that the more general question is: what is the application of mathematics? It's all just notation devices. :-) – Bob Jarvis - Reinstate Monica Aug 18 '14 at 19:02

Applications of matrices:

1. Matrix (aka quantum) Mechanics, obviously

2. Mechanics of deformable solids (where matrices describe stresses)

3. Statics (most in engineering contexts), where matrices describe stresses.

4. Symmetries (where matrices describe rotations/scaling/translations etc..)

5. Coordinate transformations, where matrices describe the transformation a coordinate system undergoes.

6. Represantation of (Linear) Operators (related to quantum mechanics but not only)

Determinants:

1. Measure volumes (in transformations etc..)

2. Measure volumes in general sense as measure (for example in Path-integral formulation, in many cases the result is expressed as a determinant of a genearally infinite-dimensional matrix)

• All of the above can be done WITHOUT matrices/determinants.(e.g. coordinate transformations are nothing but partial derivatives) But thanks anyways....!!! – GRrocks Aug 18 '14 at 16:30
• @physicslover, yes but in effect you re-derive them using a different name. One can also not use Least Action principle but use Euler-Lagrange equations, which is same thing in other guise – Nikos M. Aug 18 '14 at 16:32
• @physicslover, in case you downvoted, i dont see how this answer is foundamentaly different than the AcuriousMind's answer about Lie groups (essentially about symmetry), one can do wihout Lie groups (in essense will re-derive them in other guise) – Nikos M. Aug 18 '14 at 16:34
• @physicslover, if this is what you want to ask i suggest change the title and content of your question. In same vain complex numbers are not intrinsically needed one can very well use 2 real numbers, hope this is clear to you – Nikos M. Aug 18 '14 at 16:37
• sorry I couldn't come back earlier, and I totally agree with you, and that is precisely my question: they are simply different ways to do the same thing(in a fancier way, if you will)...And by the way, I didn't downvote :) – GRrocks Aug 19 '14 at 15:44

Lie groups are fundamental for talking about anything related to symmetries in physics on a level of some rigor, and every finite-dimensional Lie group is a matrix group. Consequently, the trace as a basic matrix operation shows up anywhere where invariance on the adjoint action of the group is needed, and the matrices are everywhere.

The Slater determinant is what multi-fermion wave-functions are, and this is not a notational trick, since that wave function is actually the n-fold wedge product of basis vectors on some space, which is (up to normalisation) also what the determinant really is.

• THanxxxxxx!!! A lot :) – GRrocks Aug 18 '14 at 16:16
• can you please suggest a few sources to study this in detail??? – GRrocks Aug 18 '14 at 16:31
• Slaterdeterminants are very clumsy compared to the product of second quantization creation operators, though. – Per Arve Aug 18 '14 at 19:19
• @PerArve: In general context, this is true, yet they can appear e.g. in the large $N$ limit of Yang-Mills theories, providing a link between representation theory and free fermions. In such contexts, they are actually quite enlightening (though still somewhat ugly, granted). – ACuriousMind Aug 18 '14 at 19:24
• @physicslover math.stackexchange.com/questions/461029/… – user5402 Aug 18 '14 at 20:58

More use of matrices:

The moment of inertia tensor needed to describe the rotational motion of rigid bodies

The Pauli matrices for spinn-1/2 (but that example is perhaps included in the Lie group example already mentioned).

Surprised nobody mentioned optics, so I will. Matrices are used extensively in geometric optics and polarization.

Examples include the ABCD matrix method for representing the effects of optical elements (e.g lenses and mirrors) in ray optics, and the Mueller matrices/Stokes vectors to represent the effects of polarizers and plates on the polarization of light.

Other than notation devices ...

Matrices are far more than mere notational devices, but even if they were, don't deride notational devices! More compact notation simplifies writing, simplifies reading, simplifies thinking, and because of that, it enables new ways of thinking.

Think of the progress from marks on a stick to representing numbers by devices such as MMXIV to place value notation such as 2014. Even those marks on a stick were extremely important notational devices. Our inborn number sense is limited to $7\pm 2$. Beyond nine, it's just "many". Those marks on a stick were important in letting our human ancestors herd sheep and cattle, knowing when to plant and when to harvest. Notational devices oftentimes are the key that lets mathematics and humanity advance to the next plateau. Zero? That's just a notational device, literally so in the case of the zero in 2014. Abstract symbols like "1", "4", and "8"? Those are nothing more than notational devices for |, ||||, and |||||||||. The symbols "+", "-", etc.: Those are just notational devices, too. We could be verbose and circumlocutious and use words and Roman numerals instead. Writing 2014+342 is just a bunch of notational devices.

Conceptually, we could do even complex mathematics using nothing but marks on a stick and the right words. Except we couldn't. Notational devices are important. They let us see the forest for the trees. In the case of matrices, I don't see how physicists could have developed quantum mechanics or how NASA could have gotten to the Moon without them. Quantum mechanics uses infinite dimension matrices. Writing that down using marks on a stick would take a long time. The Kalman filter, while not infinite dimension, is hairy in its own right. The matrix notation is what led to the development of to the signal processing concepts that in turn led to the Kalman filter.