# Applications of singular matricies

1. Jan 21, 2006

### quasi426

Are there any applications that always involve the use of noninvertible or singular matrices?? I know there are plenty for invertible ones. Thanks.

2. Jan 21, 2006

### George Jones

Staff Emeritus
If A is a matrix, then the eigenvalues a of A make the matrix A - aI singular.

Also, projections.

There must must be many more applications.

Regards,
George

3. Jan 21, 2006

### mathwonk

the concept of cohomology is a group that is the kernel of some usually geometrically meaningful linear map.

having cohomology means the map is not invertible, i.e. ahs a kernel.

for example (homology) there is a "boundary" map from (linear combinations of) paths to linear combinations of points, taking each path to the difference of its endpoints, hence closed paths are characterized by having boundary equal to zero.

thus the space of closed paths is the kernel of the boundary map. now there is also a boundary map from (linear combinations of) parametrized surfaces, to (linear combinations of) paths, taking each surface to its boundary path. moreover all boundaries oif surfaces are closed paths.

thus boundaries of surfaces are a subgroup of the closed loops. the quotient of these two spaces is called the (singualr) homology of the space.

in de rham cohomology, the derivative map takes a differential form to d of it. in dimensions 1, 2, and 3, the exterior drivative is called variously "gradient", "curl", and "divergence", for physical reasons. It is an easy calculation to show that the divergence of a curl is zero as is the curlk of a gradient.

tyhe extebnt to which the converse holds measures exactly the same topology as the homology groups. and is called the cohomology groups.

in sheaf cohomology, tyhe psace of globals, ections of a oine buyndle is the kernel of a coboundary map as well. the representation of all lineat bundles of the same dergee allows one to represent all their cohomology groups as the kernel;s of a family of matrices.

the coboundary map having a kernel is equivalent to the line bundlke having a section, so the singualr matrices paramewtrize the most interesting line bundkles those having sections.

this is mumford's approach to riemann's theta fucntion. hence the theta divisor on a jacobian varietya ssociated to a riemann surface has a determinantal equation, realizing it as the pullback of the family of singualr matrices of a certain dimension.

the list goes on and on. in almost all of mathematics, a problem is studied by defining an obstruction to that problem having a solution. thus the obstruction amp is a linear map from the space of all problems to som e other space, and the kernel of this map is the space of solvable problems,.....