Polar Decompositions: Purpose of sqrt(T*T)

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SUMMARY

The equation T=Ssqrt(T*T) illustrates the purpose of the square root in polar decompositions, particularly in the context of complex numbers. In the 1x1 case, T can be expressed as T=re^(it), where t=argT and r=|T|, with |T| being represented as sqrt(T*T). This relationship highlights that the polar decomposition extends from the 1x1 case to n-dimensional matrices, maintaining the property that a unitary matrix has an absolute value of 1, represented as e^(it).

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About the equation T=Ssqrt(T*T), what purpose does the sqrt(T*T) serve?
 
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Think about the 1x1 case, where T is a complex number. The usual polar decomposition you learn in high school tells you that T=re^(it), where t=argT and r=|T|. But the absolute value of T is just sqrt(T*T) in this case. (Also notice that a unitary 1x1 matrix is necessarily something of absolute value 1, i.e. something of the form e^(it).) So the nxn polar decomposition is just a generalization of this.
 
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