Ytukyg

You could think of a determinant as a volume. Think of the columns of the matrix as vectors at the origin forming the edges of a skewed box. The determinant gives the volume of that box. For example, in 2 dimensions, the columns of the matrix are the edges of a rhombus.

You can derive the algebraic properties from this geometrical interpretation. For example, if two of the columns are linearly dependent, your box is missing a dimension and so it's been flattened to have zero volume.

In addition to the answers, above, the determinant is a function from the set of square matrices into the real numbers that preserves the operation of multiplication:
begin{equation}det(AB) = det(A)det(B) end{equation}
and so it carries $some$ information about square matrices into the much more familiar set of real numbers.

Some examples:

The determinant function maps the identity matrix $I$ to the identity element of the real numbers ($det(I) = 1$.)

Which real number does not have a multiplicative inverse? The number 0. So which square matrices do not have multiplicative inverses? Those which are mapped to 0 by the determinant function.

What is the determinant of the inverse of a matrix? The inverse of the determinant, of course. (Etc.)

This "operation preserving" property of the determinant explains some of the value of the determinant function and provides a certain level of "intuition" for me in working with matrices.

Here is a recording of my lecture on the geometric definition of determinants:

Geometric definition of determinants

It has elements from the answers by Jamie Banks and John Cook, and goes into details in a leisurely manner.

I too find the way determinants are treated in exterior algebra most intuitive. The definition is given on page 46 of Landsberg's "Tensors: Geometry and Applications". Two examples below will tell you everything you need to know.

Say, you are give a matrix
$$A=begin{pmatrix}a&b\c&dend{pmatrix}$$
and asked to compute its determinant. You can think of the matrix as a linear operator $f:mathbb R^2tomathbb R^2$ defined by

$$begin{pmatrix}x\yend{pmatrix}mapstobegin{pmatrix}a&b\c&dend{pmatrix}begin{pmatrix}x\yend{pmatrix}.$$

If you define the standard basis vector by $e_1=begin{pmatrix}1\0end{pmatrix}$ and $e_2=begin{pmatrix}0\1end{pmatrix}$, you can then define $f$ by the values it assumes on the basis vectors: $f(e_1)=ae_1+ce_2$ and $f(e_2)=be_1+de_2$.

The linear operator $f$ is extended to bivectors by
$$f(e_1wedge e_2)=f(e_1)wedge f(e_2).$$

Then you can write

$$f(e_1wedge e_2)=(ae_1+ce_2)wedge(be_1+de_2)=(ad-bc)e_1wedge e_2,$$

where I used distributivity and anticommutativity of the wedge product (the latter implies $awedge a=0$ for any vector $a$). So, we get the determinant as a scalar factor in the above equation, that is

$$f(e_1wedge e_2)=det(A),e_1wedge e_2.$$

The same procedure works for 3-by-3 matrices, you just need to use a trivector. Say, you are given $$B=begin{pmatrix}a_{11}&a_{12}&a_{13}\a_{21}&a_{22}&a_{23}\a_{31}&a_{32}&a_{33}end{pmatrix}.$$

It defines a linear operator $g:mathbb R^3to mathbb R^3$

$$begin{pmatrix}x\y\zend{pmatrix}mapsto begin{pmatrix}a_{11}&a_{12}&a_{13}\a_{21}&a_{22}&a_{23}\a_{31}&a_{32}&a_{33}end{pmatrix} begin{pmatrix}x\y\zend{pmatrix},$$

for which we have

$$g(e_1)=a_{11}e_1+a_{21}e_2+a_{31}e_3,quad g(e_2)=a_{12}e_1+a_{22}e_2+a_{32}e_3,quad g(e_3)=a_{13}e_1+a_{23}e_2+a_{33}e_3$$

on the standard basis $e_1=begin{pmatrix}1\0\0end{pmatrix}$, $e_2=begin{pmatrix}0\1\0end{pmatrix}$, $e_3=begin{pmatrix}0\0\1end{pmatrix}$. The operator $g$ is extended to trivectors by
$$g(e_1wedge e_2wedge e_3)=g(e_1)wedge g(e_2)wedge g(e_3),$$

which gives

$$g(e_1wedge e_2wedge e_3)=(a_{11}e_1+a_{21}e_2+a_{31}e_3)wedge(a_{12}e_1+a_{22}e_2+a_{32}e_3)wedge(a_{13}e_1+a_{23}e_2+a_{33}e_3).$$

If you then follows the rules of $wedge$ such as distributivity, anticommutativity, and associativity, you get
$$g(e_1wedge e_2wedge e_3)=det(B), e_1wedge e_2wedge e_3.$$

It works in exactly the same way in higher dimensions.

The top exterior power of an $n$-dimensional vector space $V$ is one-dimensional. Its elements are sometimes called pseudoscalars, and they represent oriented $n$-dimensional volume elements.

A linear operator $f$ on $V$ can be extended to a linear map on the exterior algebra according to the rules $f(alpha) = alpha$ for $alpha$ a scalar and $f(A wedge B) = f(A) wedge f(B), f(A + B) = f(A) + f(B)$ for $A$ and $B$ blades of arbitrary grade. Trivia: some authors call this extension an outermorphism. The extended map will be grade-preserving; that is, if $A$ is a homogeneous element of the exterior algebra of grade $m$, then $f(A)$ will also have grade $m$. (This can be verified from the properties of the extended map I just listed.)

All this implies that a linear map on the exterior algebra of $V$ once restricted to the top exterior power reduces to multiplication by a constant: the determinant of the original linear transformation. Since pseudoscalars represent oriented volume elements, this means that the determinant is precisely the factor by which the map scales oriented volumes.

For the record I'll try to give a reply to this old question, since I think some elements can be added to what has been already said.

Even though they are basically just (complicated) expressions, determinants can be mysterious when first encountered. Questions that arise naturally are: (1) how are they defined in general?, (2) what are their important properties?, (3) why do they exist?, (4) why should we care?, and (5) why does their expression get so huge for large matrices?

Since $2times2$ and $3times3$ determinants are easily defined explicitly, question (1) can wait. While (2) has many answers, the most important ones are, to me: determinants detect (by becoming 0) the linear dependence of $n$ vectors in dimension $n$, and they are an expression in the coordinates of those vectors (rather than for instance an algorithm). If you have a family of vectors that depend (or at least one of them depends) on a parameter, and you need to know for which parameter values they are linearly dependent, than trying to use for instance Gaussian elimination to detect linear dependence can run into trouble: one might need assumptions on the parameter to assure some coefficient is nonzero, and even then dividing by it gives very messy expressions. Provided the number of vectors equals the dimension $n$ of the space, taking a determinant will however immediately transform the question into an equation for the parameter (which one may or may not be capable of solving, but that is another matter). This is exactly how one obtains an equation in eigenvalue problems, in case you've seen those. This provides a first answer to (4). (But there is a lot more you can do with determinants once you get used to them.)

As for question (3), the mystery of why determinants exist in the first place can be reduced by considering the situation where one has $n-1$ given linearly independent vectors, and asks when a final unknown vector $vec x$ will remain independent from them, in terms of its coordinates. The answer is that it usually will, in fact always unless $vec x$ happens to be in the linear span $S$ of those $n-1$ vectors, which is a subspace of dimension $n-1$. For instance, if $n=2$ (with one vector $vec v$ given) the answer is "unless $vec x$ is a scalar multiple of $vec v$". Now if one imagines a fixed (nonzero) linear combination of the coordinates of $vec x$ (the technical term is a linear form on the space), then it will become $0$ precisely when $vec x$ is in some subspace of dimension $n-1$. With some luck, this can be arranged to be precisely the linear span $S$. (In fact no luck is involved: if one extends the $n-1$ vectors by one more vector to a basis, then expressing $vec x$ in that basis and taking its final coordinate will define such a linear form; however you can ignore this argument unless you are particularly suspicious.) Now the crucial observation is that not only does such a linear combination exist, its coefficients can be taken to be expressions in the coordinates of our $n-1$ vectors. For instance in the case $n=2$ if one puts $vec v={achoose b}$ and $vec x={x_1choose x_2}$, then the linear combination $-bx_1+ax_2$ does the job (it becomes 0 precisely when $vec x$ is a scalar multiple of $vec v$), and $-b$ and $a$ are clearly expressions in the coordinates of $vec v$. In fact they are linear expressions. For $n=3$ with two given vectors, the expressions for the coefficients of the linear combination are more complicated, but they can still be explicitly written down (each coefficient is the difference of two products of coordinates, one form each vector). These expressions are linear in each of the vectors, if the other one is fixed.

Thus one arrives at the notion of a multilinear expression (or form). The determinant is in fact a multilinear form: an expression that depends on $n$ vectors, and is linear in each of them taken individually (fixing the other vectors to arbitrary values). This means it is a sum of terms, each of which is the product of a coefficient, and of one coordinate each of all the $n$ vectors. But even ignoring the coefficients, there are many such terms possible: a whopping $n^n$ of them!

However, we want an expression that becomes $0$ when the vectors are linearly dependent. Now the magic (sort of) is that even the seemingly much weaker requirement that the expression becomes $0$ when two successive vectors among the $n$ are equal will assure this, and it will moreover almost force the form of our expression upon us. Multilinear forms that satisfy this requirement are called alternating. I'll skip the (easy) arguments, but an alternating form cannot involve terms that take the same coordinate of any two different vectors, and they must change sign whenever one interchanges the role of two vectors (in particular they cannot be symmetric with respect to the vectors, even though the notion of linear dependence is symmetric; note that already $-bx_1+ax_2$ is not symmetric with respect to interchange of $(a,b)$ and $(x_1,x_2)$). Thus any one term must involve each of the $n$ coordinates once, but not necessarily in order: it applies a permutation of the coordinates $1,2,ldots,n$ to the successive vectors. Moreover, if a term involves one such permutation, then any term obtained by interchanging two positions in the permutation must also occur, with an opposite coefficient. But any two permutations can be transformed into one another by repeatedly interchanging two positions; so if there are any terms at all, then there must be terms for all $n!$ permutations, and their coefficients are all equal or opposite. This explains question (5), why the determinant is such a huge expression when $n$ is large.

Finally the fact that determinants exist turns out to be directly related to the fact that signs can be associated to all permutations in such a way that interchanging entries always changes the sign, which is part of the answer to question (3).
As for question (1), we can now say that the determinant is uniquely determined by being an $n$-linear alternating expression in the entries of $n$ column vectors, which contains a term consisting of the product of their coordinates $1,2,ldots,n$ in that order (the diagonal term) with coefficient $+1$. The explicit expression is a sum over all $n!$ permutations, the corresponding term being obtained by applying those coordinates in permuted order, and with the sign of the permutation as coefficient. A lot more can be said about question (2), but I'll stop here.

If you have a matrix

• $H$
  then you can calculate the correlationmatrix with


• $G = H times H^H$
  (H^H denotes the complex conjugated and transposed version of $H$).

If you do a eigenvalue decomposition of $G$ you get eigenvalues $lambda$ and eigenvectors $v$, that in combination $lambdatimes v$ describes the same space.

Now there is the following equation, saying:

• Determinant($H*H^H$) = Product of all eigenvalues $lambda$

I.e., if you have a $3times3$ matrix $H$ then $G$ is $3times3$ too giving us three eigenvalues.
The product of these eigenvalues give as the volume of a cuboid.
With every extra dimension/eigenvalue the cuboid gets an extra dimension.

There are excellent answers here that are very detailed.

Here I provide a simpler answer, also discussed in wikipedia. Think of the determinant as the area (in 2D; in 3D it would be the volume, etc.) of the parallelogram made by the vectors:

Keep in mind that the area of a parallelogram is the base $times$ height. Doing some tricks with the dot product, this yields the determinant:

$$
begin{vmatrix}
a & b \
c & d
end{vmatrix}
= ad - bc = Area_{parallelogram}
$$

You can place the unit vectors for each dimension to test the identity matrix by seeing that:
$$
begin{vmatrix}
1 & 0 \
0 & 1
end{vmatrix}
= ad - bc = 1 times 1 - 0 times 0 = 1 $$

This is a volume in more than a 2 by 2 matrix and will be equal to 1 in all cases as the off diagonal elements remove any effect from the only value contributing to the volume as the diagonal product of 1s. It is understood in some contexts that the coordinate system is unmodified.

(I considered making this a comment, but I thought it might deserve more attention than a comment would receive. Upvotes and downvotes will tell if I am right or wrong).

Complement about the sign of the determinant

I loved the accepted answer by Jamie, but I was frustrated that it did not give more explanation about the sign of the determinant and the notion of "rotation" or "orientation" of a vector. The answer from Marc Van Leeuwen comments more on this, but maybe not enough for everyone -- at least for me -- to understand what it means for a matrix to change the orientation of the space it transforms. So I googled the issue and ended up on the following explanation which I find excellent and accessible:

http://mathinsight.org/determinant_linear_transformation#lintrans3D

Think about a scalar equation,
$$ax = b$$
where we want to solve for $x$. We know we can always solve the equation if $aneq 0$, however, if $a=0$ then the answer is "it depends". If $bneq 0$, then we cannot solve it, however, if $b=0$ then there are many solutions (i.e. $x in mathbb{R}$). The key point is that the ability to solve the equation unambiguously depends on whether $a=0$.

When we consider the similar equation for matrices

$$mathbf{Ax} = mathbf{b}$$

the question as to whether we can solve it is not so easily settled by whether $mathbf{A}=mathbf{0}$ because $mathbf{A}$ could consist of all non-zero elements and still not be solvable for $mathbf{b}neqmathbf{0}$. In fact, for two different vectors $mathbf{y}_1 neq mathbf{0}$ and $mathbf{y}_2neq mathbf{0}$ we could very well have that

$$mathbf{Ay}_1 neq mathbf{0}$$
and
$$mathbf{Ay}_2 = mathbf{0}.$$

If we think of $mathbf{y}$ as a vector, then there are some directions in which $mathbf{A}$ behaves like non-zero (this is called the row space) and other directions where $mathbf{A}$ behaves like zero (this is called the null space). The bottom line is that if $mathbf{A}$ behaves like zero in some directions, then the answer to the question "is $mathbf{Ax} = mathbf{b}$ generally solvable for any $mathbf{b}$?" is "it depends on $mathbf{b}$". More specifically, if $mathbf{b}$ is in the column space of $mathbf{A}$, then there is a solution.

So is there a way that we can tell whether $mathbf{A}$ behaves like zero in some directions? Yes, it is the determinant! If $det(mathbf{A})neq 0$ then $mathbf{Ax} = mathbf{b}$ always has a solution. However if, $det(mathbf{A}) = 0$ then $mathbf{Ax} = mathbf{b}$ may or may not have a solution depending on $mathbf{b}$ and if there is one, then there are an infinite number of solutions.

One way to treat define the determinant which makes clear the relation between all the various notions you mentioned is as follows:

Given a vector space $E$ of dimmension $n$ over the field $K$ and a basis $B=(b_1,...,b_n)$ of $E$, the determinant is the unique (nonzero) alternating multilinear $n$-form $phi$ of $E$ which satisfies $phi(b_1,...,b_n)=1$.

This simply means that the determinant is a function $phi$ which takes a tuple $(x_1,...,x_n)$ of $n$ vectors of $E$ and returns a scalar from the field $K$, such that

(1) $phi$ is linear in each of the $n$ variables $(x_1,...,x_n)$ (it's "multilinear")

(2) if two of the $x_i$'s are equal, then $phi(x_1,...,x_n)=0$ ($phi$ is "alternating")

(3) It turns out that the set of functions $phi$ satisfying the two above properties are all multiples of eachother. So we choose a basis $B$ of $e$ and say that the determinant is the function $phi$ satisfying the above properties which maps $B$ to $1$.

Of course it's not immediately obvious that such a function $phi$ exists and is unique!

To simplify slightly we will take the vector space $E$ to be $K^n$ and the basis $B$ to be the canonical basis.

It turns out that the determinant satisfies the miraculous property that $det(x_1,...,x_n) neq 0$ if and only if $(x_1,...,x_n)$ is a basis.

Now... given $n$ vectors $x_1,...,x_n$ such that for the coordinates in the basis $B$ of $x_i$ are $(a_{i,1},...,a_{i,n})$, the determinant of the $n$-vectors $x_1,...,x_n$ can be shown to be equal to

$sum_{sigma in S_n} sgn(sigma)a_{1,sigma(1)}...a_{n,sigma(n)}$

which should be familiar to you as the expression for the determinant in terms of permutations. Here $S_n$ is the symmetric group, i.e the set of permutations of ${1,2,..,n}$ and $sgn(sigma)$ is the signature of the permutation $sigma$.

To make the link between the determinant of a set of $n$ vectors to the determinant of a matrix, just note that the matrix $A=(a_{i,j})$ is exactly them matrix whose column vectors are $x_1,...,x_n$.

Thus when we take the determinant of a matrix, what we are really doing is evaluating a function in terms of the $n$ column vectors. We said earlier that this function is nonzero if and only if the $n$ vectors form a basis - in other words, if and only if the matrix is of full rank, i.e iff its invertible.

So the abstract definition the determinant as a function which maps a set of vectors to the scalar field (while obeying some nice properties like linearity) is equivalent to a function from matrices to the scalar field which is nonzero exactly when the matrix is invertible. Moreover, this function turns out to be multiplicative! (Consequently, the restriction of this function to the set of invertible matrices gives is a group homomorphism from $(Gl_n(K), times)$ to $(K/{0},*)$.

The expression of the determinant of a matrix in terms of permutations can be used to derive many of the nice properties you are familiar with, for example

• a matrix and its transpose have the same det




• det of a triangular matrix is the product of the diagonal elements




• the Laplace-formula a.k.a cofactor expansion which tells you how to calculate the determinant in terms of a weighted sum of determinants of submatrices:

$det(A)=sum_{i=1}^{n}(-1)^{i+j}a_{i,j}Delta_{i,j}$

where $Delta_{i,j}$ is the determinant of the matrix obtained from $A$ by removing the row $i$ and the column $j$, known as the minor $(i,j)$.

Imagine a completely general system of equations

$$a_{11} x_1 + a_{12} x_2 + a_{13} x_3 = b_1$$
$$a_{21} x_1 + a_{22} x_2 + a_{23} x_3 = b_2$$
$$a_{31} x_1 + a_{32} x_2 + a_{33} x_3 = b_3$$

If we solve for the variables $x_i$ in terms of the other variables and write the results in lowest terms, we'll see that the expressions for each $x_i$ all have the same functions of $a_{ij}$ in the denominator. (Say that we work over the integers.) This expression is (up to a unit) the determinant of the system.

If you pick some systematic way of solving $n times n$ systems, say Gaussian elimination, you can use it to crank out a formula for this determinant.

I think this is a lot more natural than the other approaches because you start with something straightforward and common like a system of linear equations, then you put your head down and solve it, and out pops this notion.

Of course this only gives you the answer up to a sign, but this actually makes sense, because there's an arbitrary choice of sign going in.

Garibaldi has a paper that presents this approach and some related ones, entitled The determinant and characteristic polynomial are not ad hoc constructions. (To formalize this you want to bring in a little ring theory so that you have formal notions of indeterminates and so forth.)