en>Mark viking: Added context to first sentence

2013-09-06T19:53:05Z

Added context to first sentence

← Older revision		Revision as of 21:53, 6 September 2013
Line 1:		Line 1:
	~~Claude~~ is ~~her title~~ and ~~she completely digs~~ that ~~title~~. ~~Climbing~~ is ~~what love doing~~. ~~I am a cashier~~ and I'll be ~~promoted quickly~~. ~~Years ago we moved to Kansas~~.<br><br>~~Stop by my website~~ [~~http~~://~~www~~.~~Myccos.com~~/~~UserProfile/tabid/61/userId/430/Default.aspx extended car warranty~~]		{{Unreferenced\|date=November 2006}}
			In [[mathematics]], an '''involutory matrix''' is a [[matrix (mathematics)\|matrix]] that is its own inverse. That is, matrix '''A''' is an involution if and only if '''A'''<sup>2</sup> = '''I'''.
			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.

			Involutory matrices are all [[square root of a matrix\|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. If '''A''' is an ''n × n'' matrix, then '''A''' is involutory if and only if ½('''A''' + '''I''') is [[Idempotent_matrix\|idempotent]].

			An involutory matrix which is also [[symmetric matrix\|symmetric]] is an [[orthogonal matrix]], and thus represents an [[isometry]] (a linear transformation which preserves [[Euclidean distance]]). A [[Reflection (linear algebra)\|reflection]] matrix is an example of an involutory matrix.

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

			==Examples==
			Some simple examples of involutory matrices are shown below.

			:<math>
			\begin{array}{cc}
			\mathbf{I}=\begin{pmatrix}
			1 & 0 & 0 \\
			0 & 1 & 0 \\
			0 & 0 & 1
			\end{pmatrix}
			; &
			\mathbf{I}^{-1}=\begin{pmatrix}
			1 & 0 & 0 \\
			0 & 1 & 0 \\
			0 & 0 & 1
			\end{pmatrix}
			\\
			\\
			\mathbf{R}=\begin{pmatrix}
			1 & 0 & 0 \\
			0 & 0 & 1 \\
			0 & 1 & 0
			\end{pmatrix}
			; &
			\mathbf{R}^{-1}=\begin{pmatrix}
			1 & 0 & 0 \\
			0 & 0 & 1 \\
			0 & 1 & 0
			\end{pmatrix}
			\\
			\\
			\mathbf{S}=\begin{pmatrix}
			+1 & 0 & 0 \\
			0 & -1 & 0 \\
			0 & 0 & -1
			\end{pmatrix}
			; &
			\mathbf{S}^{-1}=\begin{pmatrix}
			+1 & 0 & 0 \\
			0 & -1 & 0 \\
			0 & 0 & -1
			\end{pmatrix}
			\\
			\end{array}
			</math>

			where

			:'''I''' is the [[identity matrix]] (which is trivially involutory);
			:'''R''' is an identity matrix with a pair of interchanged rows;
			:'''S''' is a [[signature matrix]].

			An interesting general condition exists, for 2 × 2 matrices, where any matrix that may be written in the form '''A''' or '''A'''<sup>T</sup> below:

			:<math>\mathbf{A}=\begin{pmatrix} a & b \\ \frac{(1-a^2)}{b} & -a \end{pmatrix};\quad \mathbf{A}^\mathrm{T}=\begin{pmatrix} a & \frac{(1-a^2)}{b} \\ b & -a \end{pmatrix}</math>

			is involutory.

			For example, for a matrix '''M''' of this form, where we set ''a'' = 1, ''b'' = 1, we have

			:<math>\mathbf{M}=\begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix};\quad\Longrightarrow\quad
			\mathbf{M}^2=\begin{pmatrix}
			1\times 1+1\times 0 & 1\times 1+1\times -1 \\ 0\times 1-1\times 0 & 0\times 1-1\times -1 \end{pmatrix}
			=\begin{pmatrix}
			1 & 0 \\ 0 & 1 \end{pmatrix} = \mathbf{I}
			</math>

			==See also==
			*[[Involution (mathematics)]]
			*[[Idempotence]]
			*[[Affine involution]]

			{{DEFAULTSORT:Involutory Matrix}}
			[[Category:Matrices]]

en>Citation bot: Updating page numbers after recent improvement to Template:Cite book. Removed redundant parameters.

2008-12-16T04:29:01Z

Updating page numbers after recent improvement to Template:Cite book. Removed redundant parameters.

New page

Claude is her title and she completely digs that title. Climbing is what love doing. I am a cashier and I'll be promoted quickly. Years ago we moved to Kansas.<br><br>Stop by my website [http://www.Myccos.com/UserProfile/tabid/61/userId/430/Default.aspx extended car warranty]

First-order reduction - Revision history

en>Mark viking: Added context to first sentence

en>Citation bot: Updating page numbers after recent improvement to Template:Cite book. Removed redundant parameters.