About the equation T=Ssqrt(T*T), what purpose does the sqrt(T*T) serve?
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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