مدونة المهندس العربي: Square Root of Matrix A

An n × n matrix A is diagonalizable if there is a matrix V and a diagonal matrix D such that

A = V D V - 1

. This happens if and only if A has n eigenvectors which constitute a basis for Cⁿ. In this case, V can be chosen to be the matrix with the n eigenvectors as columns, and a square root of

A

$R = V S V^{-1}, \,$

where S is any square root of D. Indeed,

$(V D^{1/2} V^{-1})^2 = V D^{1/2} (V^{-1} V) D^{1/2} V^{-1} = V D V^{-1} = A\,$

For example, the matrix $A = \bigl(\begin{smallmatrix}\\ 33&24\\ 48&57\end{smallmatrix} \bigr)$ can be diagonalized as

V D V - 1

, where $V = \bigl( \begin{smallmatrix}\\ 1&~\;1\\ 2&-1\end{smallmatrix} \bigr)$ and $D = \bigl( \begin{smallmatrix}\\ 81&0\\ ~\;0&9\end{smallmatrix} \bigr)$ .

D

has principal square root $D^{1/2} = \bigl( \begin{smallmatrix}\\ 9&0\\ 0&3\end{smallmatrix} \bigr)$ , giving the square root $A^{1/2} = V D^{1/2} V^{-1} = \bigl( \begin{smallmatrix}\\ 5&2\\ 4&7\end{smallmatrix} \bigr)$

مدونة المهندس العربي

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السبت، 3 مارس 2012

Square Root of Matrix A

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