Introduction Fact Although not all matrices are invertible,when an inverse exists,it is unique. The inverse is unique. Proof. ●AECnxn AB=BA=In,where BeCnxn 色电有这大习 Matrix Theory Inverse -8/35
Introduction Fact Although not all matrices are invertible, when an inverse exists, it is unique. ⇒ The inverse is unique. Proof. A ∈ C n×n AB = BA = In, where B ∈ C n×n AC = CA = In, where C ∈ C n×n Then, B = BIn = B(AC) = (BA)C = InC = C. Matrix Theory Inverse - 8/35
Introduction Fact Although not all matrices are invertible,when an inverse exists,it is unique. The inverse is unique. Proof. ●AECnxn AB=BA=In,where BeCnxn oAC=CA=I,where CECnxn 色电年这k习 Matrix Theory Inverse -8/35
Introduction Fact Although not all matrices are invertible, when an inverse exists, it is unique. ⇒ The inverse is unique. Proof. A ∈ C n×n AB = BA = In, where B ∈ C n×n AC = CA = In, where C ∈ C n×n Then, B = BIn = B(AC) = (BA)C = InC = C. Matrix Theory Inverse - 8/35
Introduction Fact Although not all matrices are invertible,when an inverse exists,it is unique. The inverse is unique. Proof. ●AECnxn AB=BA=In,where BE Cnxn oAC=CA=In,where CECnxn 色电年这k习 Matrix Theory Inverse -8/35
Introduction Fact Although not all matrices are invertible, when an inverse exists, it is unique. ⇒ The inverse is unique. Proof. A ∈ C n×n AB = BA = In, where B ∈ C n×n AC = CA = In, where C ∈ C n×n Then, B = BIn = B(AC) = (BA)C = InC = C. Matrix Theory Inverse - 8/35
Introduction Fact Although not all matrices are invertible,when an inverse exists,it is unique. The inverse is unique. Proof. ●AE Cnxn AB=BA=In,where BCnxn oAC=CA=In,where CECnxn Then, B 色电年这大习 Matrix Theory Inverse -8/35
Introduction Fact Although not all matrices are invertible, when an inverse exists, it is unique. ⇒ The inverse is unique. Proof. A ∈ C n×n AB = BA = In, where B ∈ C n×n AC = CA = In, where C ∈ C n×n Then, B = BIn = B(AC) = (BA)C = InC = C. Matrix Theory Inverse - 8/35
Introduction Fact Although not all matrices are invertible,when an inverse exists,it is unique. The inverse is unique. Proof. ●AE Cnxn ●AB=BA=In,where B∈Cnxn oAC=CA=In,where CECnxn Then, B=BIn 色电年这大习 Matrix Theory Inverse -8/35
Introduction Fact Although not all matrices are invertible, when an inverse exists, it is unique. ⇒ The inverse is unique. Proof. A ∈ C n×n AB = BA = In, where B ∈ C n×n AC = CA = In, where C ∈ C n×n Then, B = BIn = B(AC) = (BA)C = InC = C. Matrix Theory Inverse - 8/35