 | Determinant: Encyclopedia II - Determinant - Properties
Determinant - Properties
The determinant is a multiplicative map in the sense that
for all n-by-n matrices A and B.
This is generalized by the Cauchy-Binet formula to products of non-square matrices.
It is easy to see that and thus
for all n-by-n matrices A and all scalars r.
A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have
.
Expressed differently: the vectors v1,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero.
A matrix and its transpose have the same determinant:
.
The determinants of a complex matrix and of its conjugate transpose are conjugate:
.
(Note the conjugate transpose is identical to the transpose for a real matrix)
If A and B are similar, i.e., if there exists an invertible matrix X such that A = X − 1BX, then by the multiplicative property,
.
This means that the determinant is a similarity invariant. Because of this, the determinant of some linear transformation T : V → V for some finite dimensional vector space V is independent of the basis for V. The relationship is one-way, however: there exist matrices which have the same determinant but are not similar.
If A is a square n-by-n matrix with real or complex entries and if λ1,...,λn are the (complex) eigenvalues of A listed according to their algebraic multiplicities, then
.
This follows from the fact that A is always similar to its Jordan normal form, an upper triangular matrix with the eigenvalues on the main diagonal.
From this connection between the determinant and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:
.
Performing the substitution in the above equation yields
.
Determinant - Derivative
The determinant of real square matrices is a polynomial function from to , and as such is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:
where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have
or, more colloquially,
if the entries in the matrix X are sufficiently small. The special case where A is equal to the identity matrix I yields
.
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 Adapted from the Wikipedia article "Properties", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
