怎麼證明rank(A^tA)=rank(A)

Question

怎麼證明rank(A^tA)=rank(A)
怎麼證明rank(A^tA)=rank(A)

Answer 1

A為一m*n矩陣,證明rank(ATA)=rank(A)先證N(ATA)=N(A)令x€Rn,x€N(ATA),=>ATA(x)=0=>xTATAx=0=>||Ax||2=0=>Ax=0=>x€N(A)∴N(ATA)包含於N(A)又令x€N(A)=>Ax=0=>AT(Ax)=0=>x€N(ATA)∴N(A)包含於N(ATA)∴N(A)=N(ATA)∴dim(Rn)=n=nullity(A)+rank(A)=nullity(ATA)+rank(ATA)又nullity(A)=nullity(ATA)∴rank(A)=rank(ATA)