Outline (Level 1-2) ○Linear Algebra o Notations and Fundamental Concepts o Matrix Multiplication o Operational Properties o Linear Space o Quadratic Form and Positive Definite Matrix o Matrix Calculus 5/130
Outline (Level 1-2) 1 Linear Algebra Notations and Fundamental Concepts Matrix Multiplication Operational Properties Linear Space Quadratic Form and Positive Definite Matrix Matrix Calculus 5 / 130
Outline (Level 2-3) Notations and Fundamental Concepts o Matrix and Vector 6/130
Outline (Level 2-3) Notations and Fundamental Concepts Matrix and Vector 6 / 130
1.1.Notations and Fundamental Concepts 1.1.1.Matrix and Vector Matrix and Transpose:A E Rx",m rows,n columns d11 a12 d13 ain d21 a22 a23 a2n A= aml am2 am3 amn d11 a21 a31 aml AT= d12 a22 a32 am2 ain a2n a3n amn 7/130
1.1. Notations and Fundamental Concepts 1.1.1. Matrix and Vector Matrix and Transpose: A ∈ R m×n , m rows, n columns A = a11 a12 a13 · · · a1n a21 a22 a23 · · · a2n . . . . . . . . . . . . . . . am1 am2 am3 · · · amn A T = a11 a21 a31 · · · am1 a12 a22 a32 · · · am2 . . . . . . . . . . . . . . . a1n a2n a3n · · · amn 7 / 130
Vector and Matrix:d-dimensional column vector x and its transpose x'or x: X1 X2 X= ,x=X=(1,x2,…,xa Xd Denote the j-th column of A by aj,4..or A.,j,a is column vector. A=[a1,a2,..,an. Denote the i-th row of A by af,4i.:or 4i.,af is row vector.Can be understand as the row version of the i-th column of 4 A= a】 8/130
Vector and Matrix: d-dimensional column vector x and its transpose x ′ or x T : x = x1 x2 . . . xd , x T = x ′ = (x1, x2, · · · , xd) ▶ Denote the j-th column of A by aj , A:, j or A∗, j , aj is column vector. A = [a1, a2, · · · , an] . ▶ Denote the i-th row of A by a T i , Ai,: or Ai,∗, a T i is row vector. Can be understand as the row version of the i-th column of A T A = a T 1 a T 2 . . . a T m 8 / 130
Outline (Level 1-2) ○Linear Algebra o Notations and Fundamental Concepts Matrix Multiplication o Operational Properties o Linear Space o Quadratic Form and Positive Definite Matrix o Matrix Calculus 9/130
Outline (Level 1-2) 1 Linear Algebra Notations and Fundamental Concepts Matrix Multiplication Operational Properties Linear Space Quadratic Form and Positive Definite Matrix Matrix Calculus 9 / 130