《机器学习与知识发现》教学资源:Linear Algebra Review and Reference
We use the notation aii (or Ai,Aii,etc)to denote the entry of A in the ith row and jth column: a11 012··· ain a21 a22 a2n A= aml am2 amn We denote the ith column of A by aj or A.j: … We denote the ith row of A by af or Ai.:: A= Note that these definitions are ambiguous (for example,the a and af in the previous two definitions are not the same vector).Usually the meaning of the notation should be obvious from its use. 2 Matrix Multiplication The product of two matrices AE Rmxn and BERnxp is the matrix C=AB∈RmxP, where C,=∑AkB k=1 Note that in order for the matrix product to exist,the number of columns in A must equal the number of rows in B.There are many ways of looking at matrix multiplication,and we'll start by examining a few special cases. 3• We use the notation aij (or Aij , Ai,j , etc) to denote the entry of A in the ith row and jth column: A =      a11 a12 · · · a1n a21 a22 · · · a2n . . . . . . . . . . . . am1 am2 · · · amn      . • We denote the jth column of A by aj or A:,j : A =   | | | a1 a2 · · · an | | |  . • We denote the ith row of A by a T i or Ai,: : A =      — a T 1 — — a T 2 — . . . — a T m —      . • Note that these definitions are ambiguous (for example, the a1 and a T 1 in the previous two definitions are not the same vector). Usually the meaning of the notation should be obvious from its use. 2 Matrix Multiplication The product of two matrices A ∈ R m×n and B ∈ R n×p is the matrix C = AB ∈ R m×p , where Cij = Xn k=1 AikBkj . Note that in order for the matrix product to exist, the number of columns in A must equal the number of rows in B. There are many ways of looking at matrix multiplication, and we’ll start by examining a few special cases. 3
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