电子科技大学：《矩阵理论 Matrix Theory》课程教学资源（课件讲稿）08 Unitary Matrices

Matrix Theory -Special Matrices Again 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 Special Matrices Again 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 ––Special Matrices Again School of Mathematical Sciences Teaching Group Matrix Theory Special Matrices Again

Unitary and Orthogonal Matrices Outline Unitary and Orthogonal Matrices Application 1:Givens rotation Application 2:Permutation matrix Discovery Journey References 奇电有头子 Matrix Theory Special Matrices -2/23

Unitary and Orthogonal Matrices Outline Unitary and Orthogonal Matrices Application 1: Givens rotation Application 2: Permutation matrix Discovery Journey References Matrix Theory Special Matrices - 2/23

Unitary and Orthogonal Matrices Definitions 奇电有头 Matrix Theory S Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AA H = A HA = I. ▸ orthogonal if AA T = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23

Unitary and Orthogonal Matrices Definitions A matrix A Cnxn is 奇电有这头子 Matrix Theory Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AA H = A HA = I. ▸ orthogonal if AA T = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23

Unitary and Orthogonal Matrices Definitions A matrix A Cnxn is ·unitary if AAH=AHA=1. 命电有这女子 Matrix Theory Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AAH = A HA = I. ▸ orthogonal if AA T = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23

Unitary and Orthogonal Matrices Definitions A matrix Ae Cnxn is ·unitary if AAH=AHA=1. orthogonal if AA=AA=I. 命电有这女 Matrix Theory Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AAH = A HA = I. ▸ orthogonal if AAT = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23

Unitary and Orthogonal Matrices Definitions A matrix Ae Cnxn is unitary if AAH=AHA=1. orthogonal if AA=AA=1. Remarks: 奇电有这头 Matrix Theory Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AAH = A HA = I. ▸ orthogonal if AAT = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23

Unitary and Orthogonal Matrices Definitions A matrix Ae Cnxn is unitary if AAH=AHA=1. orthogonal if AA=AA=1. Remarks: A unitary matrix:columns (or rows)constitute an orthonormal basis for C. 奇电有头 Matrix Theory Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AAH = A HA = I. ▸ orthogonal if AAT = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23

Unitary and Orthogonal Matrices Definitions A matrix Ae Cnxn is unitary if AAH=AHA=1. ·orthogonal if AA=AA=1. Remarks: A unitary matrix:columns (or rows)constitute an orthonormal basis for C. A orthogonal matrix:columns (or rows)constitute an orthonormal basis for R. 命电有这女子 Matrix Theory Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AAH = A HA = I. ▸ orthogonal if AAT = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23

Unitary and Orthogonal Matrices Definitions A matrix Ae Cnxn is unitary if AAH=AHA=1. ·orthogonal if AA=AA=1. Remarks: A unitary matrix:columns (or rows)constitute an orthonormal basis for C. A orthogonal matrix:columns (or rows)constitute an orthonormal basis for R. The identity matrix is orthogonal as well as unitary. 务这头子 Matrix Theory Special Matrices -3/23

Unitary and Orthogonal Matrices Definitions A matrix A ∈ C n×n is ▸ unitary if AAH = A HA = I. ▸ orthogonal if AAT = A T A = I. Remarks: ▸ A unitary matrix: columns (or rows) constitute an orthonormal basis for C n . ▸ A orthogonal matrix: columns (or rows) constitute an orthonormal basis for R n . ▸ The identity matrix is orthogonal as well as unitary. Matrix Theory Special Matrices - 3/23