P is an orthogonal matrix and A is a periodic | vedclass

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(A) Given $Q = PAP^T$ and $P$ is an orthogonal matrix,so $P^TP = PP^T = I$.We need to find $X = P^TQ^{2005}P$.First,calculate $Q^2 = (PAP^T)(PAP^T) =

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(A) Given $Q = PAP^T$ and $P$ is an orthogonal matrix,so $P^TP = PP^T = I$.We need to find $X = P^TQ^{2005}P$.First,calculate $Q^2 = (PAP^T)(PAP^T) = PA(P^TP)AP^T = PA(I)AP^T = PA^2P^T$.By induction,$Q^n = PA^nP^T$.Thus,$Q^{2005} = PA^{2005}P^T$.Substituting this into the expression for $X$:$X = P^T(PA^{2005}P^T)P = (P^TP)A^{2005}(P^TP) = I \cdot A^{2005} \cdot I = A^{2005}$.Since $A$ is a periodic matrix with period $4$,$A^{k+1} = A$ where $k=4$. Thus $A^5 = A$.We can write $2005 = 4 	imes 501 + 1$.Therefore,$A^{2005} = A^{4 	imes 501 + 1} = (A^4)^{501} \cdot A = I^{501} \cdot A = A$.Hence,$X = A$.

$P$ is an orthogonal matrix and $A$ is a periodic matrix with period $4$,and $Q = PAP^T$. Then $X = P^TQ^{2005}P$ will be equal to

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