A*a + i = a

Dec 2009
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Russia
Prove that there not such matrix A in M_(nxn)(C) so that:
A*A + I = A.

(When A The conjugate transpose of the complex matrix A)



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Opalg

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Aug 2007
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Leeds, UK
Prove that there not such matrix A in M_(nxn)(C) so that:
A*A + I = A.

(When A The conjugate transpose of the complex matrix A)
Notice that \(\displaystyle A^* = (A^*A+I)^* = A^*A+I = A\). Therefore \(\displaystyle A^2-A+I=0\) and it follows that each eigenvalue of A must satisfy the equation \(\displaystyle \lambda^2-\lambda+1=0\). But that equation does not have real roots, which contradicts the fact that \(\displaystyle A^*=A\).
 
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