电子科技大学：《矩阵理论 Matrix Theory》课程教学资源（课件讲稿）02 Matrices Intro

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

MATRICES ARE UBIQUITOUS in computer science statistics applied mathematics Motivated largely by technological developments that generate extremely large scientific and Internet datasets,recent years have witnessed exciting developments in the theory and practice of matrix algorithms. 色电这女了 Matrix Theory Matrices -213

MATRICES ARE UBIQUITOUS in ▸ computer science ▸ statistics ▸ applied mathematics Motivated largely by technological developments that generate extremely large scientific and Internet datasets, recent years have witnessed exciting developments in the theory and practice of matrix algorithms. Matrix Theory Matrices - 2/13

What is a Matrix? Outline What is a Matrix? Some special types of matrix from nonzero pattern 奇电有头子 Matrix Theory Matrices -3/13

What is a Matrix? Outline What is a Matrix? Some special types of matrix from nonzero pattern Matrix Theory Matrices - 3/13

What is a Matrix? A matrix definition An array of numbers a11 ain A aml amn with m rows and n columns is an m x n matrix. 奇电有这头子 Matrix Theory Matrices -4/13

What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13

What is a Matrix? A matrix definition An array of numbers a11 a1n aml ..amn with m rows and n columns is an m x n matrix. Element aij is located in position(i,j). 命电有这女子 Matrix Theory Matrices -4/13

What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13

What is a Matrix? A matrix definition An array of numbers a11 a1n amn with m rows and n columns is an m x n matrix. Element aij is located in position (i,j). The element aij are scalars,namely,real or complex numbers. 命电有这女子 Matrix Theory Matrices -4/13

What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13

What is a Matrix? A matrix definition An array of numbers a11 A= amn with m rows and n columns is an m x n matrix. Element aij is located in position(i,j). The element aij are scalars,namely,real or complex numbers. .The set of real numbers is R;the set of complex numbers is C. 命电有这女 Matrix Theory Matrices -4/13

What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13

What is a Matrix? A matrix definition An array of numbers a11 A- aml amn with m rows and n columns is an m x n matrix. Element aij is located in position(i,j). The element aij are scalars,namely,real or complex numbers. The set of real numbers is R;the set of complex numbers is C. Remarks: 奇电有这头 Matrix Theory Matrices -4/13

What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13

What is a Matrix? A matrix definition An array of numbers a11 a1n A aml amn with m rows and n columns is an m x n matrix. Element aij is located in position(i,j). The element aij are scalars,namely,real or complex numbers. The set of real numbers is R;the set of complex numbers is C. Remarks: 1.A matrix is a rectangular pattern of numbers-we usually enclose the numbers with brackets. 命电有这女 Matrix Theory Matrices -4/13

What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13

What is a Matrix? A matrix definition An array of numbers a11 。。 a1n aml amn with m rows and n columns is an m x n matrix. Element aij is located in position (i,j). The element aij are scalars,namely,real or complex numbers. The set of real numbers is R;the set of complex numbers is C. Remarks: 1.A matrix is a rectangular pattern of numbers-we usually enclose the numbers with brackets. 2.These elements can be any numbers we choose-positive, negative,zero,fractions,decimals,and so on. 奇电这头了 Matrix Theory Matrices -4/13

What is a Matrix? A matrix definition An array of numbers A = ⎛ ⎜ ⎝ a11 . . . a1n ⋮ ⋮ am1 . . . amn ⎞ ⎟ ⎠ with m rows and n columns is an m × n matrix. ▸ Element aij is located in position (i, j). ▸ The element aij are scalars, namely, real or complex numbers. ▸ The set of real numbers is R; the set of complex numbers is C. Remarks: 1. A matrix is a rectangular pattern of numbers –we usually enclose the numbers with brackets. 2. These elements can be any numbers we choose - positive, negative, zero, fractions, decimals, and so on. 3. To refer briefly to a specific matrix we might label it, usually with a capital letter. Matrix Theory Matrices - 4/13