Outline (Level 1-2) ○Linear Algebra o Notations and Fundamental Concepts o Matrix Multiplication Operational Properties o Linear Space o Quadratic Form and Positive Definite Matrix o Matrix Calculus 15/130
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 15 / 130
1.3.Operational Properties Identity matrix: 1eem,amd与={0i打 1 i=j Diagonal Matrix D=diag (du.da.),and D d i=j Matrix transpose theorem: (4) A (AB)T = BTA (A+B) :AT+BT 16/130
1.3. Operational Properties ▶ Identity matrix： I ∈ R n×n , and Iij = ( 1 i = j 0 i ̸= j ▶ Diagonal Matrix D = diag (d1, d2, · · · , dn), and Dij = ( di i = j 0 i ̸= j ▶ Matrix transpose theorem: (A T ) T = A (AB) T = B TA T (A + B) T = A T + B T 16 / 130
Outline (Level 2-3) Operational Properties ●Symmetric Matrix o Trace of Matrix o Vector Norm 17/130
Outline (Level 2-3) Operational Properties Symmetric Matrix Trace of Matrix Vector Norm 17 / 130
1.3.1.Symmetric Matrix A E R"x",if A =A,then A is a symmetric matrix. if 4=-47,then A is a skew symmetric (or antisymmetric or antimetric) matrix. For any matrix, o symmetric matrix:A+7,A7A.A47 o skew symmetric matrix:A-A7 Any matrix is a sum of symmetric matrix and skew symmetric matrix A=2M+A0+5M-A0 Denote the set of all symmetric matrices as S" 18/130
1.3.1. Symmetric Matrix ▶ A ∈ R n×n , if A = A T , then A is a symmetric matrix. ▶ if A = −A T , then A is a skew symmetric (or antisymmetric or antimetric) matrix. ▶ For any matrix, symmetric matrix： A + A T , A TA, AAT skew symmetric matrix： A − A T ▶ Any matrix is a sum of symmetric matrix and skew symmetric matrix. A = 1 2 (A + A T ) + 1 2 (A − A T ) ▶ Denote the set of all symmetric matrices as S n 18 / 130
Outline (Level 2-3) Operational Properties o Symmetric Matrix Trace of Matrix o Vector Norm 19/130
Outline (Level 2-3) Operational Properties Symmetric Matrix Trace of Matrix Vector Norm 19 / 130