Transcription of SIMILAR MATRICES Similar Matrices - Mathematics
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SIMILAR MATRICES Similar Matrices - Mathematics
SIMILAR MatricesFix a linear transformationT:Rn Rnand an ordered basis,B= (~v1,..,~vn), ofRn. The standard matrix [T] and thematrix ofTwith respect toB, [T]B, are related by[T]S=S[T]Band [T] =S[T]BS 1and [T]B=S 1[T] hereS=[~v1| |~vn]is the change of basis matrix of the basis. In order to understand this relationshipbetter, it is convenient to take it as a definition and then study it twon nmatrices,AandB, we say thatAissimilartoBif there existsan invertiblen nmatrix,S, so thatAS= , this is equivalently, toA=SBS 1orB=S : IfAis SIMILAR toIn, thenA= :A=[2 31 2]is similarB=[100 1]. To see this, letS=[3 11 1],and computeAS=[3 11 1]= : IfAis SIMILAR toB, thenA2is SIMILAR toB2. To see this observe,that, by definition,A=SBS 1for some invertibleS.
are similar to diagonal matrices are extremely useful for computing large powers of the matrix. As such, it is natural to ask when a given matrix is similar to a diagonal matrix. We have the following complete answer: Theorem 3.1. A matrix Ais similar to a diagonal matrix if and only if there is an ordered basis B= (~v
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