[線代] normal matrix

[cholauda (cholauda)]

請教大大

A, B are nxn normal matrices, i.e. A^*A= AA^*; B^*B=BB^*,
then AB & BA have the same singular values

這該如何證明？
懇請解惑，謝謝！

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◆ From: 140.113.150.197
→ contrav     :U^*B^*A^*VV^*ABU=U^*B^*AVV^*A^*BU
→ contrav     :=V^*A^*BUU^*B^*AV=V^*A^*B^*UU^*BAV
→ cholauda    :請教大大~第一行到第二行是根據什麼呢？謝謝
※ 編輯: cholauda        來自: 140.113.150.197      (12/08 17:40)
→ contrav     :就A^*A= AA^* A和A^*互換而已 U,V是unitary VV^*=I
→ contrav     :另外第二個等號因為對角矩陣可以任意交換
→ cholauda    :感謝大大~另外不只是對角矩陣~前後根本就相等~