Transpose and Conjugate Transpose Properties of Transposition For real matrices,the conjugate transpose and the transpose are identical.That is,if AeRmx,then AH =AT. Transposing a matrix twice gives back the original, (AT)T=A,(AH)H=A. Transposition does not affect a scalar,while conjugate transposition conjugates the scalar, (AA)T=XAT,(AA)H=入AH 命电有这女子 Matrix Theory Matrix Transposition and Related .7/30
Transpose and Conjugate Transpose Properties of Transposition ▸ For real matrices, the conjugate transpose and the transpose are identical. That is, if A ∈ R m×n , then A H = A T . ▸ Transposing a matrix twice gives back the original, (A T ) T = A, (A H ) H = A. ▸ Transposition does not affect a scalar, while conjugate transposition conjugates the scalar, (λA) T = λA T , (λA) H = λA H . ▸ The transpose of a sum is the sum of the transposes, (A + B) T = A T + B T , (A + B) H = A H + B H . ▸ The transpose of a product is the product of the transposes with the factors in reverse order, (AB) T = B T A T , (AB) H = B HA H . Matrix Theory Matrix Transposition and Related - 7/30
Transpose and Conjugate Transpose Properties of Transposition For real matrices,the conjugate transpose and the transpose are identical.That is,if AeRmx,then AH =AT. Transposing a matrix twice gives back the original, (AT)=A,(AH)H=A. Transposition does not affect a scalar,while conjugate transposition conjugates the scalar, (AA)T=λAT,(AA)H=XAH The transpose of a sum is the sum of the transposes, (A+B)=A+BT,(A+B)H=AH+BH. 命电有这女子 Matrix Theory Matrix Transposition and Related -7/30
Transpose and Conjugate Transpose Properties of Transposition ▸ For real matrices, the conjugate transpose and the transpose are identical. That is, if A ∈ R m×n , then A H = A T . ▸ Transposing a matrix twice gives back the original, (A T ) T = A, (A H ) H = A. ▸ Transposition does not affect a scalar, while conjugate transposition conjugates the scalar, (λA) T = λA T , (λA) H = λA H . ▸ The transpose of a sum is the sum of the transposes, (A + B) T = A T + B T , (A + B) H = A H + B H . ▸ The transpose of a product is the product of the transposes with the factors in reverse order, (AB) T = B T A T , (AB) H = B HA H . Matrix Theory Matrix Transposition and Related - 7/30
Transpose and Conjugate Transpose Properties of Transposition For real matrices,the conjugate transpose and the transpose are identical.That is,if AeRmx,then AH=AT. Transposing a matrix twice gives back the original, (AT)=A.(AH)H=A. Transposition does not affect a scalar,while conjugate transposition conjugates the scalar, (XA)=XAT,(AA)H=TAH The transpose of a sum is the sum of the transposes, (A+B)=A+B,(A+B)H=AH+BH. The transpose of a product is the product of the transposes with the factors in reverse order, (AB)T=BTAT,(AB)H=BHAH. Matrix Theory Matrix Transposition and Related -7/30
Transpose and Conjugate Transpose Properties of Transposition ▸ For real matrices, the conjugate transpose and the transpose are identical. That is, if A ∈ R m×n , then A H = A T . ▸ Transposing a matrix twice gives back the original, (A T ) T = A, (A H ) H = A. ▸ Transposition does not affect a scalar, while conjugate transposition conjugates the scalar, (λA) T = λA T , (λA) H = λA H . ▸ The transpose of a sum is the sum of the transposes, (A + B) T = A T + B T , (A + B) H = A H + B H . ▸ The transpose of a product is the product of the transposes with the factors in reverse order, (AB) T = B T A T , (AB) H = B HA H . Matrix Theory Matrix Transposition and Related - 7/30
Transpose and Conjugate Transpose Example (AB)T=BTAT,(AB)H=BHAH 命电有这女子 Matrix Theory Matrix Transposition and Related -8/30
Transpose and Conjugate Transpose Example (AB) T = B T A T , (AB) H = B HA H . ▸ Why do we have to reverse the order of the factors when the transpose is pulled inside the product AB? ▸ Why isn’t (AB) T = A T B T ? Matrix Theory Matrix Transposition and Related - 8/30
Transpose and Conjugate Transpose Example (AB)T=BTAT, (AB)H=BHAH. Why do we have to reverse the order of the factors when the transpose is pulled inside the product AB? 命电有这女子 Matrix Theory Matrix Transposition and Related -8/30
Transpose and Conjugate Transpose Example (AB) T = B T A T , (AB) H = B HA H . ▸ Why do we have to reverse the order of the factors when the transpose is pulled inside the product AB? ▸ Why isn’t (AB) T = A T B T ? Matrix Theory Matrix Transposition and Related - 8/30