[工学]Lecture 2 Basic Concepts on Matrix.ppt

* This process is called Gram-Schmidt orthonormalization. * * * * * * * * classical adjoint ) matrix by * * * * * 映像 * * * * * CONCLUSION Basic concepts: matrix, entries, diagonal/identity/lower/upper/nonnegtive/positive matrix symmetric/skew-symmetric/Hermitian matrix, scalar multiplication, sum, product, transpose, adjoint, column space, row space, inverse, linear system, solution set, range, row operation, RREF, augmented matrix, rank, consistent, null space, nonsingular, inner product, angle, orthogonal, orthonormal, orthocomplement, determinant, minor, adjugate, quasi-triangular, quasi-diagonal, linear transformation, basis representation. * Important principles: *AC=BC doesn’t means A=B; *Every matrix has a symmetric part and a skew-symmetric part. *Range of a matrix is the column space of it; *Ax=b have a solution iff b belongs to the column space of A. *r(A)=r(AT)=r(AAT); *dim[null(A)]=n–r(A)=n-dim[range(A)]; *Every rank of 1 matrix has the form A=xyT; *A is nonsingular iff A is invertible iff Ax=b has unique solution for every b. *AB=I, then BA=I; *Every basis can be orthonormalized; *detA=∏aii for triangular matrix; *detA=∏detAii. *Every linear transformation has a basis representation as a matrix. * HOMEWORK 1. 2. 3. 4. * Lecture 2: Basic Concepts on Matrix MATRIX ANALYSIS @ HITSZ TIME: Autumn 2011 INSTRUCTOR: You-Hua Fan Reading assignment on the textbook Section 0.2-0.10 * * * Hermitian矩阵是实对称矩阵 的推广，共轭转置等于本身的矩阵A=A共轭转置 例如 1 2i 3+i -2i 5 6 3-i 6 4 * * * * * * * * * * We define the linear system for the n unknowns x1, . . . , xn to be The solution set is defined to be the subset of Rn of vectors x1, . . . , xn that satisfy each of the m equations of the system. Then the system (*) can be written as Ax=b or: * * Each operation can be realized by left matrix multiplication: * * * * Definition 2.3.2. When Ax = b has a solution we say the system is consistent. * * * * * * b * * Remarks. If AB=I, then B is the unique inverse