By Jean Gallier
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Additional info for Applications of Scientific Computation [Lecture notes]
X y Remark: Unless we are in R2 , the sign of this angle is not determined. 3, we defined the product of an m × n matrix A by a column vector x ∈ Rn . The result is a vector y ∈ Rm such that y = Ax. It follows that the m × n matrix A defines a function from Rn to Rm . On input x ∈ Rn , this function produces the output y = Ax ∈ Rm . It turns out that such functions are very special: they are linear , but we will not discuss this right now. Suppose now that we have two matrices A and B, where A is a m × n matrix and B is a n × p matrix.
Furthermore, since Ak = Ek−1 Pk−1 · · · E1 P1 A and since Gaussian elimination stops for k = n, the matrix An = En−1 Pn−1 · · · E2 P2 E1 P1 A is upper-triangular. Also note that if we let M = En−1 Pn−1 · · · E2 P2 E1 P1 , then det(M ) = ±1, and det(A) = ± det(An ). The matrices P (i, k) and Ei,j;β are called elementary matrices. 1. (Gaussian Elimination) Let A be an n × n matrix (invertible or not). Then there is some invertible matrix, M , so that U = M A is upper-triangular. The pivots are all nonzero iff A is invertible.
5. (1) Given any matrices A ∈ Mm,n , B ∈ Mn,p , and C ∈ Mp,q , we have (AB)C = A(BC) Im A = A AIn = A, that is, matrix multiplication is associative and has left and right identities. (2) Given any matrices A, B ∈ Mm,n , and C, D ∈ Mn,p , for all λ ∈ R, we have (A + B)C = AC + BC A(C + D) = AC + AD (λA)C = λ(AC) A(λC) = λ(AC), We say that matrix multiplication · : Mm,n × Mn,p → Mm,p is bilinear . 5. INVERSE OF A MATRIX; SOLVING LINEAR SYSTEMS 41 (where a and b are real numbers) has a solution iff a = 0, in which case x= b = a−1 b, a where a−1 = 1/a is the inverse of a.
Applications of Scientific Computation [Lecture notes] by Jean Gallier