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Download Matrix product. Let A be an m × n matrix. If x ∈ IR is a
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Matrix product. Let A be an m × n matrix. If x ∈ IRn is a (column) vector, then n X aik xk = bi , 1 ≤ i ≤ m Ax = b : k=1 defines the matrix product function f (x) = Ax from IRn to IRm . Similarly, if B is an n × p matrix and y ∈ IRp is a (column) vector, then p X By = x : bkj yj = xk , 1 ≤ k ≤ n j=1 defines the matrix product function g(y) = By from IRp to IRn . Then the composite function f ◦ g from IRp to IRm is given by p p X n n X X X aik bkj )yj . bkj yj = ( aik f (g(y)) = A(By) = k=1 j=1 j=1 k=1 If we define the matrix product AB by n X (AB)ij = aik bkj , k=1 then we have (1) f (g(y)) = (AB)y, and so composition of matrix product functions is consistent with matrix multiplication. ~a1 ~a2 Write the matrix A as a column of rows A = . . . and the matrix B ~am as a row of columns B = b1 b2 . . . bp . The ij term in the matrix product is given by the row-column product, (AB)ij = ~ai bj , 1 ≤ i ≤ m, 1 ≤ j ≤ p. 1