7. Let A E CX. Show that there exists a unitary matrix Q such that A*...

  

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7. Let A E C"X". Show that there exists a unitary matrix Q such that A* QAQ. Problem 8 Let A E CmXn has rank n. Show that there is a factorization of A = PH, where PE Cmxn has orthonormal columns, and H E C"X" is Hermitian positive definite. If A E C"Xn, then, show that, || A - P|2 < || A - Q|2 for any unitary matrix Q. Problem 9 Let A E CmXn has rank n have a factorization A = PH as in Problem 8. Show that ||A* A – I||2 1+ || A||2 <|A – P||2 < ||A*A – I||2 1+ on Problem 10 Let A E Cnxn and o > 0. Show that o is a singular values of A if and only if the matrix A -oI -ol A* is singular. 7. Let A E C"X". Show that there exists a unitary matrix Q such that A* QAQ. Problem 8 Let A E CmXn has rank n. Show that there is a factorization of A = PH, where PE Cmxn has orthonormal columns, and H E C"X" is Hermitian positive definite. If A E C"Xn, then, show that, || A - P|2 < || A - Q|2 for any unitary matrix Q. Problem 9 Let A E CmXn has rank n have a factorization A = PH as in Problem 8. Show that ||A* A – I||2 1+ || A||2 <|A – P||2 < ||A*A – I||2 1+ on Problem 10 Let A E Cnxn and o > 0. Show that o is a singular values of A if and only if the matrix A -oI -ol A* is singular.