證明：A是正定或半正定實(shí)對(duì)稱矩陣的充要條件是A合同與對(duì)角矩陣diag(a1,a2,...,an)
其中a1,a2,...,an都是非負(fù)數(shù).
即存在可逆矩陣C,使得C'AC=diag(a1,a2,...,an)
所以A=(C')^-1diag(a1,a2,...,an)C^-1
=(C')^-1diag(√a1,√a2,...,√an)diag(√a1,√a2,...,√an)C^-1
=[diag(√a1,√a2,...,√an)C^-1]'[diag(√a1,√a2,...,√an)C^-1]
令S=diag(√a1,√a2,...,√an)C^-1
即得A=S'S