What Are the Physical Applications of Matrices?

[andonrangelov]

I start to learn about matrices and their algebra, but I am wondering what physical application they have. I know that matrices have application in optics, which is called “Matrix Optics”, but do they have other applications? Can you give different and real physical examples with matrix algebra?
Thanks

 

[Superstring]

Tensors.

 

[SteamKing]

One basic application is determining the principal moments of inertia for 3-D bodies. Matrices can be used in many areas of kinematics. Quantum mechanics also relies heavily on matrix algebra and tensors for compact expression of its equations.

 

[DragonPetter]

the Bra and Ket notation of quantum mechanics uses matrices and matrix operations extensively. Eigenvalues have significance of a system's state in energy levels and there are all kinds of other linear algebra concepts used in things like commutators and applying a hamiltonian to a system. Most importantly is that all of the rules of how these physical properties interact obey the linear algebra theorems.

Another application is in solid-state physics in describing crystal formations, and calculating distance and angles of atoms within a crystal. A crystal can be thought of as a 3D matrix of atoms.

State space descriptions of physical systems are usually kept in matrix form.

I haven't looked into this too much, but matrices are also applied in all kinds of mathematical transforms like the Fourier transform which is used in physics to describe frequency spectrums.

Ray tracing, which attempts to represent EM, acoustic, etc. waves as discrete rays, is often done as matrix operations, because a new ray is basically a translation and angle transform of its previous ray.

Then there are many other applications of matrices in engineering and computer science.

 

[andonrangelov]

Thank you very much for those examples, but with quantum mechanics I am not familiar. I just want to have more detail example, it can be in optics or other more easy understanding fields then quantum mechanics.
For example a link or a good book recommendation will serve best. Thanks again….

 

[Studiot]

Any discipline where you have systems of simultaneous (not necessarily linear) equations can be conveniently handled by tabling them as a single matrix equation.

This is used extensively in engineering fields such as

Electrical Network Analysis
Structural Analysis
Pipe Network Analysis
etc

Very often numerical methods are used and there are matrices underlying finite element or boundary element analyses.

When you come to multidimensional analysis (calculus in 3 or more dimensions) in maths you will encounter objects called Jacobians, which do the job of simple dx and dy in simple calculus (broad brush view). These have matrix form and have applications wherever you do multidimensional analysis - pure maths or physics.

go well