nLab determinant

Idea

The determinant is the (essentially unique) universal alternating multilinear map.

Definition

Preliminaries on exterior algebra

Let Vect${}_k$ be the category of vector spaces over a field $k$, and assume for the moment that the characteristic $char(k) \neq 2$. For each $j \geq 0$, let

$$sgn_j \colon S_j \to \hom(k, k)$$

be the 1-dimensional sign representation on the symmetric group $S_j$, taking each transposition $(i j)$ to $-1 \in k^\times$. We may linearly extend the sign action of $S_j$, so that $sgn$ names a (right) $k S_j$-module with underlying vector space $k$. At the same time, $S_j$ acts on the $j^{th}$ tensor product of a vector space $V$ by permuting tensor factors, giving a left $k S_j$-module structure on $V^{\otimes j}$. We define the Schur functor

$$\Lambda^j \colon Vect_k \to Vect_k$$

by the formula

$$\Lambda^j(V) = sgn_j \otimes_{k S_j} V^{\otimes j}.$$

It is called the $j^{th}$ alternating power (of $V$).

Another point of view on the alternating power is via superalgebra. For any cosmos $\mathbf{V}$ let $CMon(\mathbf{V})$ be the category of commutative monoid objects in $\mathbf{V}$. The forgetful functor $CMon(\mathbf{V}) \to \mathbf{V}$ has a left adjoint

$$\exp(V) = \sum_{n \geq 0} V^{\otimes n}/S_n$$

whose values are naturally regarded as graded by degree $n$.

This applies in particular to $\mathbf{V}$ the category of supervector spaces; if $V$ is a supervector space concentrated in odd degree, say with component $V_{odd}$, then each symmetry $\sigma: V \otimes V \to V \otimes V$ maps $v \otimes w \mapsto -w \otimes v$ for elements $v, w \in V_{odd}$. It follows that the graded component $\exp(V)_n$ is concentrated in $parity(n)$ degree, with component $\Lambda^n(V_{odd})$.

Proposition

There is a canonical natural isomorphism $\Lambda^n(V \oplus W) \cong \sum_{j + k = n} \Lambda^j(V) \otimes \Lambda^k(W)$.

Proof

Again take $\mathbf{V}$ to be the category of supervector spaces. Since the left adjoint $\exp: \mathbf{V} \to CMon(\mathbf{V})$ preserves coproducts and since the tensor product $\otimes$ of $\mathbf{V}$ provides the coproduct for commutative monoid objects, we have a natural isomorphism

$$\exp(V \oplus W) \cong \exp(V) \otimes \exp(W).$$

Examining the grade $n$ component $\exp(V \oplus W)_n$, this leads to an identification

$$\exp(V \oplus W)_n = \sum_{j + k = n} \exp(V)_j \otimes \exp(W)_k.$$

and now the result follows by considering the case where $V, W$ are concentrated in odd degree.

Corollary

If $V$ is $n$-dimensional, then $\Lambda^j(V)$ has dimension $\binom{n}{j}$. In particular, $\Lambda^n(V)$ is 1-dimensional.

Proof

By induction on dimension. If $\dim(V) = 1$, we have that $\Lambda^0(V)$ and $\Lambda^1(V)$ are $1$-dimensional, and clearly $\Lambda^n(V) = 0$ for $n \geq 2$, at least when $char(k) \neq 2$.

We then infer

$$\array{ \Lambda^j(V \oplus k) & \cong & \sum_{p + q = j} \Lambda^p(V) \otimes \Lambda^q(k) \\ & \cong & \Lambda^j(V) \oplus \Lambda^{j-1}(V) }$$

where the dimensions satisfy the same recurrence relation as for binomial coefficients: $\binom{n+1}{j} = \binom{n}{j} + \binom{n}{j-1}$.

More concretely: if $e_1, \ldots, e_n$ is a basis for $V$, then expressions of the form $e_{n_1} \otimes \ldots \otimes e_{n_j}$ form a basis for $V^{\otimes j}$. Let $e_{n_1} \wedge \ldots \wedge e_{n_j}$ denote the image of this element under the quotient map $V^{\otimes j} \to \Lambda^j(V)$. We have

$$e_{n_1} \wedge \ldots \wedge e_{n_i} \wedge e_{n_{i+1}} \wedge \ldots \wedge e_{n_j} = -e_{n_1} \wedge \ldots \wedge e_{n_{i+1}} \wedge e_{n_i} \wedge \ldots \wedge e_{n_j}$$

(consider the transposition in $S_j$ which swaps $i$ and $i+1$) and so we may take only such expressions on the left where $n_1 \lt \ldots \lt n_j$ as forming a spanning set for $\Lambda^j(V)$, and indeed these form a basis. The number of such expressions is $\binom{n}{j}$.

Remark

In the case where $char(k) = 2$, the same development may be carried out by simply decreeing that $e_{n_1} \wedge \ldots \wedge e_{n_j} = 0$ whenever $n_i = n_{i'}$ for some pair of distinct indices $i$, $i'$.

Now let $V$ be an $n$-dimensional space, and let $f \colon V \to V$ be a linear map. By the corollary, the map

$$\Lambda^n(f) \colon \Lambda^n(V) \to \Lambda^n(V),$$

being an endomorphism on a 1-dimensional space, is given by multiplying by a scalar $D(f) \in k$. It is manifestly functorial since $\Lambda^n$ is, i.e., $D(f g) = D(f) D(g)$. The quantity $D(f)$ is called the determinant of $f$.

Determinant of a matrix

We see then that if $V$ is of dimension $n$,

$$\det \colon End(V) \to k$$

is a homomorphism of multiplicative monoids; by commutativity of multiplication in $k$, we infer that

$$\det(U A U^{-1}) = \det(A)$$

for each invertible linear map $U \in GL(V)$.

If we choose a basis of $V$ so that we have an identification $End(V) \cong Mat_n(k)$, then the determinant gives a function

$$\det \colon Mat_n(k) \to k$$

or by restriction to invertible matrices with invertible determinants a function

$$\det\colon GL_n(k) \to k^*$$

that takes products of $n \times n$ matrices to products in $k$. The determinant however is of course independent of choice of basis, since any two choices are related by a change-of-basis matrix $U$, where $A$ and its transform $U A U^{-1}$ have the same determinant. The above map is furthermore natural in $k$, hence is a natural transformation $det\colon GL_n\rightarrow -^*$ from the general linear group to the group of units of a field (or more generally a ring), which are both functors from Field (or more generally Ring) to Grp.

By following the definitions above, we can give an explicit formula:

$$\det(A) \;=\; \sum_{\sigma \in S_n} sgn(\sigma) \prod_{i = 1}^n a_{i \sigma(i)} \;$$

This may equivalently be written using the Levi-Civita symbol $\epsilon$ and the Einstein summation convention as

(1)$$\det(A) \;=\; a_{1 j_1} a_{2 j_2} \cdots a_{n j_n} \, \epsilon^{j_1 j_2 \cdots j_n}$$

which in turn may be re-written more symmetrically as

(2)$$\det(A) \;=\; \frac{1}{n!} \epsilon^{ i_1 i_2 \cdots i_n } \, a_{i_1 j_1} a_{i_2 j_2} \cdots a_{i_n j_n} \, \epsilon^{j_1 j_2 \cdots j_n}$$

Properties

We work over fields of arbitrary characteristic. The determinant satisfies the following properties, which taken together uniquely characterize the determinant. Write a square matrix $A$ as a row of column vectors $(v_1, \ldots, v_n)$.

1. $\det$ is separately linear in each column vector:

   $$\det(v_1, \ldots, a v + b w, \ldots, v_n) = a\det(v_1, \ldots, v, \ldots, v_n) + b\det(v_1, \ldots, w, \ldots, v_n)$$

2. $\det(v_1, \ldots, v_n) = 0$ whenever $v_i = v_j$ for distinct $i, j$.

3. $\det(I) = 1$, where $I$ is the identity matrix.

Other properties may be worked out, starting from the explicit formula or otherwise:

• If $A$ is a diagonal matrix, then $\det(A)$ is the product of its diagonal entries.

• More generally, if $A$ is an upper (or lower) triangular matrix, then $\det(A)$ is the product of the diagonal entries.

• If $E/k$ is a field extension and $f$ is a $k$-linear map $V \to V$, then $\det(f) = \det(E \otimes_k f)$. Using the preceding properties and the Jordan normal form? of a matrix, this means that $\det(f)$ is the product of its eigenvalues (counted with multiplicity), as computed in the algebraic closure of $k$.

• If $A^t$ is the transpose of $A$, then $\det(A^t) = \det(A)$.

Cramer’s rule

A simple observation which flows from these basic properties is

Proposition

(Cramer’s Rule)

Let $v_1, \ldots, v_n$ be column vectors of dimension $n$, and suppose

$$w = \sum_j a_j v_j.$$

Then for each $i$ we have

$$a_j \det(v_1, \ldots, v_i, \ldots, v_n) = \det(v_1, \ldots, w, \ldots, v_n)$$

where $w$ occurs as the $i^{th}$ column vector on the right.

Proof

This follows straightforwardly from properties 1 and 2 above.

For instance, given a square matrix $A$ such that $\det(A) \neq 0$, and writing $A = (v_1, \ldots, v_n)$, this allows us to solve for a vector $a$ in an equation

$$A \cdot a = w$$

and we easily conclude that $A$ is invertible if $\det(A) \neq 0$.

Remark

This holds true even if we replace the field $k$ by an arbitrary commutative ring $R$, and we replace the condition $\det(A) \neq 0$ by the condition that $\det(A)$ is a unit. (The entire development given above goes through, mutatis mutandis.)

Characteristic polynomial and Cayley-Hamilton theorem

Given a linear endomorphism $f: M\to M$ of a finite rank free unital module over a commutative unital ring, one can consider the zeros of the characteristic polynomial $\det(t \cdot 1_V - f)$. The coefficients of the polynomial are the concern of the Cayley-Hamilton theorem.

Over the real numbers: volume and orientation

A useful intuition to have for determinants of real matrices is that they measure change of volume. That is, an $n \times n$ matrix with real entries will map a standard unit cube in $\mathbb{R}^n$ to a parallelpiped in $\mathbb{R}^n$ (quashed to lie in a hyperplane if the matrix is singular), and the determinant is, up to sign, the volume of the parallelpiped. It is easy to convince oneself of this in the planar case by a simple dissection of a parallelogram, rearranging the dissected pieces in the style of Euclid to form a rectangle. In algebraic terms, the dissection and rearrangement amount to applying shearing or elementary column operations to the matrix which, by the properties discussed earlier, leave the determinant unchanged. These operations transform the matrix into a diagonal matrix whose determinant is the area of the corresponding rectangle. This procedure easily generalizes to $n$ dimensions.

The sign itself is a matter of interest. An invertible transformation $f \colon V \to V$ is said to be orientation-preserving if $\det(f)$ is positive, and orientation-reversing if $\det(f)$ is negative. Orientations play an important role throughout geometry and algebraic topology, for example in the study of orientable manifolds (where the tangent bundle as $GL(n)$-bundle can be lifted to a $GL_+(n)$-bundle structure, $GL_+(n) \hookrightarrow GL(n)$ being the subgroup of matrices of positive determinant). See also KO-theory.

Finally, we include one more property of determinants which pertains to matrices with real coefficients (which works slightly more generally for matrices with coefficients in a local field):

As a polynomial in traces of powers

On the relation between determinant and trace:

If $A$ is an $n \times n$ matrix, the determinant of its exponential equals the exponential of its trace

$$\det(\exp(A)) = \exp(tr(A)) \,.$$

More generally, the determinant of $A$ is a polynomial in the traces of the powers of $A$:

For $2 \times 2$-matrices:

$$det(A) \;=\; \tfrac{1}{2}\left( tr(A)^2 - tr(A^2) \right)$$

For $3 \times 3$-matrices:

$$det(A) \;=\; \tfrac{1}{6} \left( (tr(A))^3 - 3 tr(A^2) tr(A) + tr(A^3) \right)$$

For $4 \times 4$-matrices:

$$det(A) \;=\; \tfrac{1}{24} \left( (tr(A))^4 - 6 tr(A^2)(tr(A))^2 + 3 (tr(A^2))^2 + 8 tr(A^3) tr(A) - 6 tr(A^4) \right)$$

Generally for $n \times n$-matrices (Kondratyuk-Krivoruchenko 92, appendix B):

(3)$$det(A) \;=\; \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(tr(A^l)\right)^{k_l}$$

Proof of (3)

It is enough to prove this for semisimple matrices $A$ (matrices that are diagonalizable upon passing to the algebraic closure of the ground field) because this subset of matrices is Zariski dense (using for example the nonvanishing of the discriminant of the characteristic polynomial) and the set of $A$ for which the equation holds is Zariski closed.

Thus, without loss of generality we may suppose that $A$ is diagonal with $n$ eigenvalues $\lambda_1, \ldots, \lambda_n$ along the diagonal, where the statement can be rewritten as follows. Letting $p_k = tr(A^k) = \lambda_1^k + \ldots + \lambda_n^k$, the following identity holds:

$$\prod_{i=1}^n \lambda_i = \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } p_l^{k_l}$$

This of course is just a polynomial identity, one closely related to various of the Newton identities that concern symmetric polynomials in indeterminates $x_1, \ldots, x_n$. Thus we again let $p_k = x_1^k + \ldots + x_n^k$, and define the elementary symmetric polynomials $\sigma_k = \sigma_k(x_1, \ldots, x_n)$ via the generating function identity

$$\sum_{k \geq 0} \sigma_k t^k = \prod_{i=1}^n (1 + x_i t).$$

Then we compute

$$\array{ \sum_{k \geq 0} \sigma_k t^k & = & \prod_{i=1}^n (1 + x_i t) \\ & = & \exp\left(\sum_{i=1}^n \log(1 + x_i t)\right) \\ & = & \exp\left(\sum_{i=1}^n \sum_{k \geq 1} (-1)^{k+1} \frac{x_i^k}{k} t^k \right)\\ & = & \exp\left( \sum_{k \geq 1} (-1)^{k+1} \frac{p_k}{k} t^k\right) }$$

and simply match coefficients of $t^n$ in the initial and final series expansions, where we easily compute

$$x_1 x_2 \ldots x_n = \sum_{n = k_1 + 2k_2 + \ldots + n k_n} \prod_{l=1}^n \frac1{(k_l)!} \left(\frac{p_l}{l}\right)^{k_l} (-1)^{k_l+1}$$

This completes the proof.

In terms of Berezinian integrals

see Pfaffian for the moment

Related entries

• matrix, linear algebra, exterior algebra, characteristic polynomial

• principal submatrix

• quasideterminant, Berezinian,Jacobian, Pfaffian, hafnian, Wronskian

• Dieudonné determinant

• resultant, discriminant, hyperdeterminant

• functional determinant

• determinant line, determinant line bundle, Pfaffian line bundle

• density bundle

• pseudoscalar

• Slater determinant

• permanent

References

See also

• Wikipedia, Determinant

One derivation of the formula (3) for the determinant as a polynomial in traces of powers is spelled out in appendix B of

• L. A. Kondratyuk, I. Krivoruchenko, Superconducting quark matter in $SU(2)$ colour group, Zeitschrift für Physik A Hadrons and Nuclei March 1992, Volume 344, Issue 1, pp 99–115 (doi:10.1007/BF01291027)