Adv. Appl. Cliff ord Algebras© 2014 Springer BaselDOI 10.1007/s00006-014-0455-3

On the Determinant-like Function and theVector Determinant

Abhimanyu Pallavi Sudhir

Abstract. A generalisation of the determinant to rectangular matrices,known as the determinant-like function, has its magnitude defined pre-viously. In this paper, we show that the determinant-like function is arotation of the vector determinant. We further propose that this rotationis an identity transformation and thus the determinant-like function isin fact the same as the vector determinant. From this, we derive someproperties of the determinant-like function.

Keywords. Linear algebra; clifford algebra; exterior algebra.

1. Introduction

There have been a few papers which attempt to generalise the determinantfunction to non-square matrices such as [1], [2], and [3]. Each of these papersgeneralise the determinant - like function based on different axioms, and giverise to different generalisations of the determinant. The third of these gener-alisations does not have definition for the sign or direction of the generaliseddeterminant.

This generalisation is known as the determinant - like function and itsmagnitude can be stated recursively as follows (where m = n + h):

|detlAm×n| =

√√√√n+h∑

k0=1

detl2crossk0A (1)

[3]. Here, crossj1,j2,...,jk refers to crossing out k of the rows (if m ≥ n) orcolumns (if m < n). This equation was derived through the means of theClifford Algebra C� (Rn, 0) based on the property that the exterior productof the rows/columns of a matrix has a magnitude which is equal the absolutevalue of its determinant.

Advances inApplied Cliff ord Algebras

2 A. Pallavi Sudhir

The second is the vector determinant, defined as:

�D Am×n

=n+h∑

k=1

ek detl crosskA. (2)

However, the sign or direction of the determinant-like function still re-mains completely unknown. In this paper, we propose that this determinant-like function is in fact equivalent to the vector determinant.

The motivation for studying this particular generalisation of the deter-minant function is that by definition, its absolute value is equal to the mag-nitude of the exterior product of its column vectors, and thus equal to thehypervolume of the object formed by them, i.e. it allows the study of lower-dimensional objects in higher-dimensional spaces when m > n, although thegeometric meaning is unknown when m < n. For such reasons, it is possiblethat this function has important implications.

2. The Magnitude of the Vector Determinant and theDeterminant-like Function

Notice that the magnitude of the vector-valued determinant can be given by∥∥∥∥ �D A

m×n

∥∥∥∥ =n+h∑

k=1

ek detl crosskA. (3)

The magnitude of the Right-Hand-Side of Equation (3) is in fact exactlythe same as the Right-Hand-Side of the Equation (1). We thus observe that

∥∥∥∥ �D Am×n

∥∥∥∥ =∥∥∥∥detl A

m×n

∥∥∥∥ . (4)

Equation (4) can be true if detlAm×n = Q�DAm×n where Q ∈ SO(m).We propose that Q = I, and thus:

detlA = �DA. (5)

This clearly shows that the vector determinant is a rotation of thedeterminant-like function.

3. The Properties of the Determinant-like Function

Equation (5) immediately implies that the determinant-like function satisfiesall the defining properties of the vector determinant, namely:

• The determinant-like function is linear in its columns (if m > n) or rows(if m < n).

• The determinant-like function is 0 if any of its columns (if m > n) orrows (if m < n) are linearly dependent.

• The determinant-like function is a unit vector if its columns (if m > n)or rows (if m < n) �xj = ej where ej is the jth unit vector.

On the Determinant-like Function and the Vector Determinant

Also, through Property 1, it is trivial that:For some m× n matrix Am×n, the following holds:

detl rAm×n = rn detlAm×n. (6)

4. Conclusion

We have shown that the determinant-like function is in fact a rotation ofthe vector determinant and proposed that this rotation is in fact an iden-tity transformation. Using this, we have come to a few properties of thedeterminant-like function. As the determinant-like function seems to satisfymany of the properties of the determinant, it is possible that the determinant-like function would have some important implications.

References

[1] H. Pyle, Non-Square Determinants and Multilinear Vectors. Mathematics Asso-ciation of America 35(2) (1962), 65–69.

[2] M. Radic, A Definition of Determinant of Rectangular Matrix, 1(21) (1966),321–349.

[3] A. Pallavi Sudhir, Defining the determinant-like function for m by n matricesusing the Exterior Algebra. Advances in Applied Clifford Algebras 23(4) (2013),787–792.

Abhimanyu Pallavi SudhirDhirubhai Ambani International SchoolBandra-Kurla Complex, Bandra (East)Bombay 400098Indiae-mail: abhi99.ps@gmail.com

Received: September 29, 2013.

Accepted: January 10, 2014.