文件中找不不到关系 ID 为 rid3 的图像部件. Review of LINEAR ALGEBRA I. TA: Yujia Xie CSE 6740 Computational Data Analysis. Georgia Institute of Technology

Transcription

1 文件中找不不到关系 ID 为 rid3 的图像部件 Review of LINEAR ALGEBRA I TA: Yujia Xie CSE 6740 Computational Data Analysis Georgia Institute of Technology

2 1. Notations A R $ & : a matrix with m rows and n columns, where the entries of A are real numbers. x R & : a vector with n entries. By convention, an n-dimensional vector is often thought of as a matrix with n rows and 1 column, known as a column vector. (x + : row vector) x, : the ith element of a vector x A,. : the entry of A in the ith row and jth column

3 1. Notations a. or A :,. : the jth column of A a, + or A,,: : the ith row of A

4 2. Matrix Multiplication Matrix Multiplication: If,, then the multiplication is defined as Contents 2.1 Vector-Vector Products 2.2 Matrix-Vector Products 2.3 Matrix-Matrix Products

5 2.1 Vector-Vector Products Inner/dot Product: Given Real number Outer Product: Given

6 2.2 Matrix-Vector Products Matrix-Vector Products: Given, On one hand, the ith entry of y is equal to the inner product of the ith row of A and x, On the other hand, y is a linear combination of the columns of A

7 2.2 Matrix-Vector Products It is also possible to multiply on the left by a row vector We can also express it in two obvious ways

8 2.3 Matrix-Matrix Products Matrix Multiplication: If,, then the multiplication is defined as We can view it as a set of vector-vector products

9 2.3 Matrix-Matrix Products We can view it as a set of vector-vector products

10 3. Operations and Properties Contents 3.1 The Identity Matrix and Diagonal Matrices 3.2 The Transpose 3.3 Symmetric Matrices 3.4 TheTrace 3.5 Norms 3.6 Linear Independence and Rank 3.7 The Inverse 3.8 Orthogonal Matrices 3.9 Range and Null Space of a Matrix 3.10 The Determinant

11 3.1 Identity Matrix & Diagonal Matrices Identity matrix: diagonal matrix:

12 3.2 The Transpose The transpose of a matrix: Given Property

13 3.3 Symmetric Matrices Symmetric: A square matrix is symmetric if Anti-symmetric: A square matrix A Rn n is symmetric if Property For any square matrix is symmetric is anti-symmetric The set of all symmetric matrices of size n is usually denoted as S &.

14 3.4 The Trace The trace of a square matrix : Given Property

15 3.5 Norms Norm: Norms:

16 3.6 Linear Independence and Rank Linearly Independent: A set of vectors {x 6, x 7 x & } R $ is said to be (linearly) independent if no vector can be represented as a linear combination of the remaining vectors. Conversely, if one vector belonging to the set can be represented as a linear combination of the remaining vectors, then the vectors are said to be (linearly) dependent. Rank: The column rank of a matrix A R $ & is the size of the largest subset of columns of A that constitute a linearly independent set. In the same way, the row rank is the largest number of rows of A that constitute a linearly independent set. For any matrix A R $ &, the column rank of A is equal to the row rank of A (though we will not prove this), and so both quantities are referred to collectively as the rank of A, denoted as rank(a). Property

17 3.7 The Inverse Inverse: The inverse of a square matrix A R & & is denoted A ;6, and is the unique matrix such that Invertible or non-singular: A is invertible or non-singular if A ;6 exists Non-invertible or singular: A is non-invertible or singular if A ;6 does not exist Property

18 3.8 Orthogonal Matrices Orthogonal vectors: Two vectors x, y R & are orthogonal if x + y = 0 Normalized: A vector x R & is normalized if x 7 = 1 Orthogonal matrices: A square matrix U R & & is orthogonal if all its columns are orthogonal to each other and are normalized (the columns are then referred to as being orthonormal) Property

19 3.9 Range and Nullspace of a Matrix Span: The span of a set of vectors {x 6, x 7 x & } is the set of all vectors that can be expressed as a linear combination of {x 6, x 7 x & }, i.e. Projection: Given y R $, x, R $ Range: Given A R $ & If When A contains only a single column, a R $

20 3.9 Range and Nullspace of a Matrix Nullspace: Given A R $ & Property Orthogonal complements: disjoint subsets that together span the entire space of R &, just like N(A) and R(A + ).

21 3.10 The Determinant Denote, for A R & &, A \F,\G R (&;6) (&;6) to be the matrix that results from deleting the ith row and jth column from A Determinant: The determinant of a square matrix A R & &, is a function det : R & & R, and is denoted A or det A Example

22 3.10 The Determinant 文件中找不不到关系 ID 为 rid3 的图像部件 Classical adjoint (adjoint): Given A R & & It can be shown that for any nonsingular A R & &, Property

23 文件中找不不到关系 ID 为 rid2 的图像部件 文件中找不不到关系 ID 为 rid2 的图像部件 Thanks If you have further comments or questions, please me at