determinant

Overview

The determinant is an algebraic operation that transforms asquare matrix $M$ into a scalar. This operation has many useful andimportant properties. For example, the determinant is zero if andonly the matrix $M$ is singular (no inverse exists). The determinantalso has an important geometric interpretation as the area of aparallelogram, and more generally as the volume of ahigher-dimensional parallelepiped.

The notion of determinant predates matrices and lineartransformations. Originally, the determinant was a number associatedto a system of $n$ linear equations in $n$ variables. This number“determined” whether the system possessed a unique solution. Inthis sense, two-by-two determinants were considered by Cardano at theend of the 16th century and ones of arbitrary size (see the definitionbelow) by Leibniz about 100 years later.

Definition

Let $M$ be an $n\\times n$ matrix with entries $M_{ij}$ that areelements of a given field¹¹Most scientific and geometricapplications deal with matrices made up of real or complex numbers.However, the determinant of a matrix over any field is well definedsense and has all the properties of the more conventionaldeterminant. Indeed, many properties of the determinant remainvalid for matrices with entries in a commutative ring.. Thedeterminant of $M$ , or $\\det M$ for short, is the scalar quantity

\\det M=\\left|\\begin{array}[]{cccc}M_{11}&M_{12}&\\ldots&M_{1n}\\\\M_{21}&M_{22}&\\ldots&M_{2n}\\\\\\vdots&\\vdots&\\ddots&\\vdots\\\\M_{n1}&M_{n2}&\\ldots&M_{nn}\\end{array}\\right|=\\sum_{\\pi\\in S_{n}}\\mathrm{sgn}(%\\pi)M_{1\\pi_{1}}M_{2\\pi_{2}}\\cdots M_{n\\pi_{n}}.

(1)

The index $\\pi$ in the above sum varies over all the permutations of $\\{1,\\ldots,n\\}$ (i.e., the elements of the symmetric group $S_{n}$ .)Hence, there are $n!$ terms in the defining sum of the determinant.The symbol $\\operatorname{sgn}(\\pi)$ denotes the parity of thepermutation; it is $\\pm 1$ according to whether $\\pi$ is an even orodd permutation. Using the Einstein summation convention one can alsoexpress the above definition as

\\det M=\\epsilon_{\\pi_{1}\\pi_{2}\\dots\\pi_{n}}M^{\\pi_{1}}{}_{1}M^{\\pi_{2}}{}_{2}%\\cdots M^{\\pi_{n}}{}_{n},

(2)

where we’ve raised the first index so that $M^{i}{}_{j}=M_{ij}$ , andwhere

\\epsilon_{\\pi_{1}\\dots\\pi_{n}}=\\operatorname{sgn}(\\pi)

is known as the Levi-Civita permutation symbol.

By way of example, the determinant of a $2\\times 2$ matrix is given by

\\left|\\begin{matrix}M_{11}&M_{12}\\\\M_{21}&M_{22}\\end{matrix}\\right|=M_{11}M_{22}-M_{12}M_{21},

There are six permutations of the numbers $1,2,3$ , namely

1\\overset{+}{2}3,\\;2\\overset{+}{3}1,\\;3\\overset{+}{1}2,\\;1\\overset{-}{3}2,\\;3%\\overset{-}{2}1,\\;2\\overset{-}{1}3;

the overset sign indicates the permutation’s signature. Accordingly,the $3\\times 3$ deterimant is a sum of the following $6$ terms:

\\left|\\begin{matrix}M_{11}&M_{12}&M_{13}\\\\M_{21}&M_{22}&M_{23}\\\\M_{31}&M_{32}&M_{33}\\end{matrix}\\right|=\\begin{array}[]{rr}\\\\M_{11}M_{22}M_{33}+M_{12}M_{23}M_{31}+M_{13}M_{21}M_{32}\\\\-M_{11}M_{23}M_{32}-M_{13}M_{22}M_{31}-M_{12}M_{21}M_{33}\\end{array}

Remarks and important properties

1.
The determinant operation converts matrix multiplication intoscalar multiplication;
$\\det(AB)=\\det(A)\\det(B),$
where $A, B$ are square matrices of the same size.
2.
The determinant operation is multi-linear, and anti-symmetricwith respect to the matrix’s rows and columns. See themulti-linearity attachment for more details.
3.
The determinant of a lower triangular, or an upper triangularmatrix is the product of the diagonal entries, since all the other summandsin (1) are zero.
4.
Similar matrices (http://planetmath.org/SimilarMatrix)have the same determinant. To be more precise,let $A$ and $X$ be square matrices with $X$ invertible. Then,
$\\det(XAX^{-1})=\\det(A).$
In particular, if we let $X$ be the matrix representing a change ofbasis, this shows that the determinant is independent of the basis.The same is true of the trace of a matrix. In fact, the wholecharacteristic polynomial of an endomorphism is definable withoutusing a basis or a matrix, and it turns out that the determinant andtrace are two of its coefficients.
5.
The determinant of a matrix $A$ is zero if and only if $A$ issingular; that is, if there exists a non-trivial solution to thehomogeneous equation
$A\\mathbf{x}=\\mathbf{0}.$
6.
The transposeoperation does not change the determinant:
$\\det A^{\\scriptscriptstyle\\mathrm{T}}=\\det A.$
7.
The determinant of a diagonalizable transformation is equal to the productof its eigenvalues, counted with multiplicities.
8.
The determinant is homogeneous of degree $n$ . This means that
$\\det(kM)=k^{n}\\det M,\\quad k\\;\\text{is a scalar.}$