A determinant equal to zero means that a matrix is a singular matrix. A matrix is singular if it does not have an inverse, which means it cannot be used to solve systems of linear equations.
This can be a trouble when trying to solve a system of linear equations using the matrix inverse and finding that the inverse of a singular matrix is not defined. You might conclude that “there is no solution” and stop there.
However, it is possible to find another way to solve the system of equations when the determinant is zero without using the inverse of the matrix directly.
Why can’t we find the inverse of a matrix when the determinant is equal to zero?
Let’s remind that the determinant of a matrix is a numeric value that can be calculated from the elements of the matrix. The determinant of a matrix is used in different areas of mathematics and physics to solve problems involving matrices. Its relevance with the inverse of a matrix depends on the following equivalence, where the inverse of a matrix only exists if the determinant is different from zero, as it means division by zero:
Geometrically we could interpret the determinant of a matrix as the area (for 2x2 matrices) or the volume (for 3x3 and so on…) between all the vectors. If one vector is zero or is coplanar with others then it will not contribute to the “height” of the volume and the determinant will be zero.
An analogous example of the dependent variable could be a four-legged table that can still stand even if one leg is missing. While it is technically possible for the dependent vector to have some contribution, it is not necessary as it can be replaced by a linear combination of other vectors. Each leg could be modeled as a force in a linear equation, while the equations come from Newton’s laws.
Transformation
A matrix typically transforms one set of characteristics into another. This occurs smoothly when there is a one-to-one correspondence, also known as bijective mapping. However, if the determinant of the matrix is equal to zero, it means that there is no bijective mapping.
In bijective mapping, the information is transformed from one space to another, similar to translating between languages. The original transformation can be reversed through a one-to-one mapping, similar to encoding and decoding a message, by keeping track of the initial transformation in an external table (like a Rosetta stone of transformation).
When this happens the determinant is different from zero.
In systems of equations each row represents an equation or constraint, while each column represents a variable to be mapped. For every variable, we need one equation. If there are more equations than variables, we will not be able to satisfy all of them simultaneously. On the other hand, if there are more variables than equations, we will not have enough equations to reduce the variables to a unique result.
Problems arise when the mapping is many-to-one or one-to-many, as in these cases the determinant is zero.
This is because it may not be possible to uniquely determine the original set of characteristics from the transformed set, or vice versa. In other words, the information may not be preserved or reversible under the transformation.
From many to one, could be interpreted as a projection, an encoder, or a summarization, it is a convergence process. It could give you the correlation with a vector.
> Solution. In this case to find a solution we need to define additional restrictions, mainly a cost function to optimize the best solution.
From one to many, could be interpreted as a generator or a decoder. it is a divergent process. We could generate many artistic draws with all having a similar style as the degree of freedom of the output is not higher than the input, at least in a deterministic process.
> Solution. In this case, we don’t want the best solution for data, but instead, the best system or model to describe the underlying behavior to correctly generate data with that model.
To find the desired solution at a technical level, you can apply the Moore-Penrose inverse. This method is an optimization process that minimizes mean square error/value by reducing the “many” options to a single one. To do this, you basically will need to multiply both sides of the equation by the transposed matrix and then solve as usual.
This concept may be difficult to understand, but it is easy to implement. That’s why I am providing more context and explanation rather than just writing out formulas. I hope that some of the key ideas will stay with you.
Why does this trick of multiplying by the transpose matrix work? In short, it is a projection that gets rid of non-orthogonal direction and as the closest point to the origin point is contained by an orthogonal line to the null space, then you get a minimization.
Next, I’ll provide an example of how to handle a determinant equal to zero situation where there are more columns than rows. As a specific example with a real application could be as long as this story I will write it separately.
Spoiler: It has to do with cable robots and statics equilibrium. In general for problems that are “statically indeterminate”. That is the number of unknowns is greater than the number of equations.
Reference
- Left and right pseudo-inverse formulas got from: Outline Properties on the Matrix Inverse Matrix Inverse Lemma — ppt download (slideplayer.com)
- Definition of “statically indeterminate” by 3.5 Indeterminate Loads — Engineering Mechanics: Statics (upei.ca)
- Video from 3Blue1Brown — The determinant: https://youtu.be/Ip3X9LOh2dk
