电子科技大学：《矩阵理论 Matrix Theory》课程教学资源（课件讲稿）06 Matrix Transposition and Related

Matrix Theory 净6 -Matrix Transposition and Related School of Mathematical Sciences Teaching Group Main Reference books Fuzhen Zhang.Matrix Theory-Basic Results and Techniques,Second Edition. Springer,2011. llse C.F.Ipsen,Numerical Matrix Analysis:Linear Systems and Least Squares. SlAM,2009. Reference books: Roger A.Horn and Charles A.Johnson:Matrix Analysis.Cambridge University Press,1985. Gene H.Golub and Charles F.Van Loan:Matrix Computations,Third Edition. Johns Hopkins Press,1996. Nicholas J.Higham.Accuracy and Stability of Numerical Algorithms,Second Edition.SIAM,2002. Y.Saad.Iterative Methods for Sparse Linear Systems,Second Edition.SIAM, Philadelphia,2003. Matrix Theory Matrix Transposition and Related Maintained by Yan-Fei Jing

: Main Reference books ▸ Fuzhen Zhang. Matrix Theory-Basic Results and Techniques, Second Edition. Springer, 2011. ▸ Ilse C. F. Ipsen, Numerical Matrix Analysis: Linear Systems and Least Squares. SIAM, 2009. Reference books: ▸ Roger A. Horn and Charles A. Johnson: Matrix Analysis. Cambridge University Press, 1985. ▸ Gene H. Golub and Charles F. Van Loan: Matrix Computations, Third Edition. Johns Hopkins Press, 1996. ▸ Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms, Second Edition. SIAM, 2002. ▸ Y. Saad. Iterative Methods for Sparse Linear Systems, Second Edition. SIAM, Philadelphia, 2003. Maintained by Yan-Fei Jing Matrix Theory ––Matrix Transposition and Related School of Mathematical Sciences Teaching Group Matrix Theory Matrix Transposition and Related

Transpose and Conjugate Transpose Outline Transpose and Conjugate Transpose Inner and Outer Products,Again Symmetric and Hermitian Matrices Exercises 色老这子 Matrix Theory Matrix Transposition and Related .2/30

Transpose and Conjugate Transpose Outline Transpose and Conjugate Transpose Inner and Outer Products, Again Symmetric and Hermitian Matrices Exercises Matrix Theory Matrix Transposition and Related - 2/30

Transpose and Conjugate Transpose Transpose 奇老有头子 Matrix Theory Matrix Transposition and Related -3/30

Transpose and Conjugate Transpose Transpose Matrix Theory Matrix Transposition and Related - 3/30

Transpose and Conjugate Transpose Transpose Transposing a matrix amounts to turning rows into columns and vice versa 奇老有头子 Matrix Theory Matrix Transposition and Related -3/30

Transpose and Conjugate Transpose Transpose Transposing a matrix amounts to turning rows into columns and vice versa. Matrix Theory Matrix Transposition and Related - 3/30

Transpose and Conjugate Transpose Transpose Transposing a matrix amounts to turning rows into columns and vice versa. If a11 a12 a1n A= a21 a22 a2n aml am2 amn 色电这女子 Matrix Theory Matrix Transposition and Related -3/30

Transpose and Conjugate Transpose Transpose Transposing a matrix amounts to turning rows into columns and vice versa. If A = ⎛ ⎜ ⎜ ⎜ ⎝ a11 a12 . . . a1n a21 a22 . . . a2n ⋮ ⋮ ⋮ am1 am2 . . . amn ⎞ ⎟ ⎟ ⎟ ⎠ , Matrix Theory Matrix Transposition and Related - 3/30

Transpose and Conjugate Transpose Transpose Transposing a matrix amounts to turning rows into columns and vice versa. If a11 a12 a1n A= a21 a22 a2n aml am2 amn Then its transpose AT eCnxm is obtained by converting rows to columns, a11 a21 aml AT a12 322 am2 a1n a2n amn 色电这女子 Matrix Theory Matrix Transposition and Related .3/30

Transpose and Conjugate Transpose Transpose Transposing a matrix amounts to turning rows into columns and vice versa. If A = ⎛ ⎜ ⎜ ⎜ ⎝ a11 a12 . . . a1n a21 a22 . . . a2n ⋮ ⋮ ⋮ am1 am2 . . . amn ⎞ ⎟ ⎟ ⎟ ⎠ , Then its transpose A T ∈ C n×m is obtained by converting rows to columns, A T = ⎛ ⎜ ⎜ ⎜ ⎝ a11 a21 . . . am1 a12 a22 . . . am2 ⋮ ⋮ ⋮ a1n a2n . . . amn ⎞ ⎟ ⎟ ⎟ ⎠ . Matrix Theory Matrix Transposition and Related - 3/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. 命电有这女子 Matrix Theory Matrix Transposition and Related -4/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. Matrix Theory Matrix Transposition and Related - 4/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. If Ae Cmxn is a matrix,its conjugate transpose AH(also sometimes denoted as A*)Chxm is obtained by converting rows to columns, 奇电有头子 Matrix Theory Matrix Transposition and Related -4/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. If A ∈ C m×n is a matrix, its conjugate transpose A H ( also sometimes denoted as A ∗ ) ∈ C n×m is obtained by converting rows to columns, Matrix Theory Matrix Transposition and Related - 4/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. If AeCmxn is a matrix,its conjugate transpose AH(also sometimes denoted as A*)Cnxm is obtained by converting rows to columns, and in addition,taking the complex conjugates of the elements, 命电有这女子 Matrix Theory Matrix Transposition and Related -4/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. If A ∈ C m×n is a matrix, its conjugate transpose A H ( also sometimes denoted as A ∗ ) ∈ C n×m is obtained by converting rows to columns, and in addition, taking the complex conjugates of the elements, Matrix Theory Matrix Transposition and Related - 4/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. If Ae Cmxn is a matri>ⅸ，its conjugate transpose A(also sometimes denoted as A*)ECnxm is obtained by converting rows to columns, and in addition,taking the complex conjugates of the elements, a11 a21 aml AH= a12 a22 3m2 a1n a2n 3mn 奇电有这头 Matrix Theory Matrix Transposition and Related -4/30

Transpose and Conjugate Transpose Conjugate Transpose There is a second type of transposition that requires more work when the matrix elements are complex numbers. If A ∈ C m×n is a matrix, its conjugate transpose A H ( also sometimes denoted as A ∗ ) ∈ C n×m is obtained by converting rows to columns, and in addition, taking the complex conjugates of the elements, A H = ⎛ ⎜ ⎜ ⎜ ⎝ a11 a21 . . . am1 a12 a22 . . . am2 ⋮ ⋮ ⋮ a1n a2n . . . amn ⎞ ⎟ ⎟ ⎟ ⎠ . Matrix Theory Matrix Transposition and Related - 4/30