电子科技大学：《矩阵理论 Matrix Theory》课程教学资源（课件讲稿）03 Matrices-special matrices

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

Some special types of matrix Outline Some special types of matrix Some special matrices 奇电有头子 Matrix Theory Matrices -2/14

Some special types of matrix Outline Some special types of matrix Some special matrices Matrix Theory Matrices - 2/14

Some special types of matrix Some special types of matrix 命电有这女子 Matrix Theory Matrices -3/14

Some special types of matrix Some special types of matrix ▸ zero matrix: a matrix, every element of which is zero. Note: Zero matrices of different orders are different. ▸ square matrix: as the name suggests, has the same number of rows as columns. ▸ row matrix (or vector): a 1 × m matrix, i.e., y ∈ C 1×m. ( 1 0 −1 2 ) ▸ column matrix (or vector): an n × 1 matrix, i.e., x ∈ C n×1 or shorter, x ∈ C n . ⎛ ⎜ ⎝ 6 4 3 ⎞ ⎟ ⎠ Matrix Theory Matrices - 3/14

Some special types of matrix Some special types of matrix zero matrix:a matrix,every element of which is zero. 命电有这女子 Matrix Theory Matrices -3/14

Some special types of matrix Some special types of matrix ▸ zero matrix: a matrix, every element of which is zero. Note: Zero matrices of different orders are different. ▸ square matrix: as the name suggests, has the same number of rows as columns. ▸ row matrix (or vector): a 1 × m matrix, i.e., y ∈ C 1×m. ( 1 0 −1 2 ) ▸ column matrix (or vector): an n × 1 matrix, i.e., x ∈ C n×1 or shorter, x ∈ C n . ⎛ ⎜ ⎝ 6 4 3 ⎞ ⎟ ⎠ Matrix Theory Matrices - 3/14

Some special types of matrix Some special types of matrix zero matrix:a matrix,every element of which is zero. Note:Zero matrices of different orders are different. 命电有这女子 Matrix Theory Matrices -3/14

Some special types of matrix Some special types of matrix ▸ zero matrix: a matrix, every element of which is zero. Note: Zero matrices of different orders are different. ▸ square matrix: as the name suggests, has the same number of rows as columns. ▸ row matrix (or vector): a 1 × m matrix, i.e., y ∈ C 1×m. ( 1 0 −1 2 ) ▸ column matrix (or vector): an n × 1 matrix, i.e., x ∈ C n×1 or shorter, x ∈ C n . ⎛ ⎜ ⎝ 6 4 3 ⎞ ⎟ ⎠ Matrix Theory Matrices - 3/14

Some special types of matrix Some special types of matrix zero matrix:a matrix,every element of which is zero. Note:Zero matrices of different orders are different. square matrix:as the name suggests,has the same number of rows as columns. 奇老这头子 Matrix Theory Matrices -3/14

Some special types of matrix Some special types of matrix ▸ zero matrix: a matrix, every element of which is zero. Note: Zero matrices of different orders are different. ▸ square matrix: as the name suggests, has the same number of rows as columns. ▸ row matrix (or vector): a 1 × m matrix, i.e., y ∈ C 1×m. ( 1 0 −1 2 ) ▸ column matrix (or vector): an n × 1 matrix, i.e., x ∈ C n×1 or shorter, x ∈ C n . ⎛ ⎜ ⎝ 6 4 3 ⎞ ⎟ ⎠ Matrix Theory Matrices - 3/14

Some special types of matrix Some special types of matrix zero matrix:a matrix,every element of which is zero. Note:Zero matrices of different orders are different. square matrix:as the name suggests,has the same number of rows as columns. row matrix (or vector):a 1 x m matrix,i.e.,ye Clxm. (10-12) 命电有这女子 Matrix Theory Matrices -3/14

Some special types of matrix Some special types of matrix ▸ zero matrix: a matrix, every element of which is zero. Note: Zero matrices of different orders are different. ▸ square matrix: as the name suggests, has the same number of rows as columns. ▸ row matrix (or vector): a 1 × m matrix, i.e., y ∈ C 1×m. ( 1 0 −1 2 ) ▸ column matrix (or vector): an n × 1 matrix, i.e., x ∈ C n×1 or shorter, x ∈ C n . ⎛ ⎜ ⎝ 6 4 3 ⎞ ⎟ ⎠ Matrix Theory Matrices - 3/14

Some special types of matrix Some special types of matrix zero matrix:a matrix,every element of which is zero. Note:Zero matrices of different orders are different. square matrix:as the name suggests,has the same number of rows as columns. row matrix(or vector))：小a1×m matrix,i.e,y∈Clxm. (10-12) column matrix (or vector):an nx 1 matrix,i.e.,x eCnx1 or shorter,xeC. 6 4 3 命电有这女 Matrix Theory Matrices -3/14

Some special types of matrix Some special types of matrix ▸ zero matrix: a matrix, every element of which is zero. Note: Zero matrices of different orders are different. ▸ square matrix: as the name suggests, has the same number of rows as columns. ▸ row matrix (or vector): a 1 × m matrix, i.e., y ∈ C 1×m. ( 1 0 −1 2 ) ▸ column matrix (or vector): an n × 1 matrix, i.e., x ∈ C n×1 or shorter, x ∈ C n . ⎛ ⎜ ⎝ 6 4 3 ⎞ ⎟ ⎠ Matrix Theory Matrices - 3/14

Some special types of matrix diagonal matrix:a square matrix with zeros everywhere except possibly on the diagonal which runs from the top left to the bottom right.This diagonal is called the leading diagonal. 命电有这女子 Matrix Theory Matrices -4/14

Some special types of matrix ▸ diagonal matrix: a square matrix with zeros everywhere except possibly on the diagonal which runs from the top left to the bottom right. This diagonal is called the leading diagonal. Here are some diagonal matrices: E = ( 2 0 0 −5 ) F = ⎛ ⎜ ⎝ 1 0 0 0 2 0 0 0 3 ⎞ ⎟ ⎠ G = ⎛ ⎜ ⎝ 1 0 0 0 0 0 0 0 1 ⎞ ⎟ ⎠ Note: Whereas all the non-diagonal elements are zero, the elements on the leading diagonal can be any number including zero. Matrix Theory Matrices - 4/14

Some special types of matrix diagonal matrix:a square matrix with zeros everywhere except possibly on the diagonal which runs from the top left to the bottom right.This diagonal is called the leading diagonal. Here are some diagonal matrices: 1 00 0 00 001 奇老有这头 Matrix Theory Matrices -4/14

Some special types of matrix ▸ diagonal matrix: a square matrix with zeros everywhere except possibly on the diagonal which runs from the top left to the bottom right. This diagonal is called the leading diagonal. Here are some diagonal matrices: E = ( 2 0 0 −5 ) F = ⎛ ⎜ ⎝ 1 0 0 0 2 0 0 0 3 ⎞ ⎟ ⎠ G = ⎛ ⎜ ⎝ 1 0 0 0 0 0 0 0 1 ⎞ ⎟ ⎠ Note: Whereas all the non-diagonal elements are zero, the elements on the leading diagonal can be any number including zero. Matrix Theory Matrices - 4/14