“Involutory matrix”的意思、由来-开放百科全书

In mathematics, an involutory matrix is a matrix that is its own inverse. That is, multiplication by matrix A is an involution if and only if A² = I. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any nonsingular matrix multiplied by its inverse is the identity.^[1]

Examples

One of the three classes of elementary matrix is involutory, namely the row-interchange elementary matrix. A special case of another class of elementary matrix, that which represents multiplication of a row or column by −1, is also involutory; it is in fact a trivial example of a signature matrix, all of which are involutory.

Clearly, any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks.

Symmetry

An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance). Conversely every orthogonal involutory matrix is symmetric.^[3]

Properties

If A is an n × n matrix, then A is involutory if and only if ½(A + I) is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices.^[4]

If A is an involutory matrix in M(n ,ℝ), a matrix algebra over the real numbers, then the subalgebra {x I + y A: x,y ∈ ℝ} generated by A is isomorphic to the split-complex numbers.

if A and B are two involutory matrices which commute with each other then AB is also involutory.

if A is involutory matrix then every natural power of A is involutory. In fact, Aⁿ will be equal to A if n is odd and I if n is even.

See also

References

1. ^{{citation | last = Higham | first = Nicholas J. | authorlink = Nicholas Higham | contribution = 6.11 Involutory Matrices | doi = 10.1137/1.9780898717778 | isbn = 978-0-89871-646-7 | location = Philadelphia, PA | mr = 2396439 | pages = 165–166 | publisher = Society for Industrial and Applied Mathematics (SIAM) | title = Functions of Matrices: Theory and Computation | url = https://books.google.com/books?id=2Wz_zVUEwPkC&pg=PA165 | year = 2008}}.
2. ^Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, 2nd edition, pp 12,13 Academic Press {{ISBN|0-12-435560-9}}
3. ^{{citation | last = Govaerts | first = Willy J. F. | doi = 10.1137/1.9780898719543 | isbn = 0-89871-442-7 | location = Philadelphia, PA | mr = 1736704 | page = 292 | publisher = Society for Industrial and Applied Mathematics (SIAM) | title = Numerical methods for bifurcations of dynamical equilibria | url = https://books.google.com/books?id=rqvYq19qwiwC&pg=PA292 | year = 2000}}.
4. ^¹{{citation | last = Bernstein | first = Dennis S. | contribution = 3.15 Facts on Involutory Matrices | edition = 2nd | isbn = 978-0-691-14039-1 | location = Princeton, NJ | mr = 2513751 | pages = 230–231 | publisher = Princeton University Press | title = Matrix Mathematics | url = https://books.google.com/books?id=-c0NxJg4vHMC&pg=PA230 | year = 2009}}.