Cayley-Hamilton theorem over a Field
... works for matrices over an integral domain R, since R can be embedded in a field. To me it would be surprising if some similar proof did not work even if there are zero-divisors. I try to keep this material at "advanced placement" High School level. We assume that the concept of determinant has been ...
... works for matrices over an integral domain R, since R can be embedded in a field. To me it would be surprising if some similar proof did not work even if there are zero-divisors. I try to keep this material at "advanced placement" High School level. We assume that the concept of determinant has been ...
The decompositional approach to matrix computation
... In 1961,Jamcs Wilkinson gave thcfirst back- point additions atid multiplications. T h e algoward rounding-error analysis of the solutions of rithm Cholesky proposed corresponds to thc dilinear systems." Here, the division of labor is agram in the lower right of Figurc 3. complete. IIe givcs one anal ...
... In 1961,Jamcs Wilkinson gave thcfirst back- point additions atid multiplications. T h e algoward rounding-error analysis of the solutions of rithm Cholesky proposed corresponds to thc dilinear systems." Here, the division of labor is agram in the lower right of Figurc 3. complete. IIe givcs one anal ...
A Level Maths - Further Maths FP1
... Understand successive transformations and the connection with matrix multiplication. ...
... Understand successive transformations and the connection with matrix multiplication. ...
lectures on solution of linear equations
... General strategy to solve Ax = b is to transform the system in a way that does not effect the solution but renders it easier to calculate. Let M by any nonsingular matrix and z be the solution of MAz = Mb Then z = (MA)-1 Mb = A-1 M-1 M b = A-1 b = x Call “pre-multiplying” or “multiply from the left” ...
... General strategy to solve Ax = b is to transform the system in a way that does not effect the solution but renders it easier to calculate. Let M by any nonsingular matrix and z be the solution of MAz = Mb Then z = (MA)-1 Mb = A-1 M-1 M b = A-1 b = x Call “pre-multiplying” or “multiply from the left” ...
Updated Course Outline - Trinity College Dublin
... [ Sydsaeter, ch. 6 & 7 (up to 7.9) ] [ Chiang, ch. 6, 7.1 to 7.3, 8.1, 8.5 (up to p. 199), 9.3 & 9.5 ] 1. Definition and interpretation a. Difference quotient b. Derivative c. Increasing and decreasing functions d. Limits e. Continuity vs differentiability 2. Rules of Differentiation a. Constant fun ...
... [ Sydsaeter, ch. 6 & 7 (up to 7.9) ] [ Chiang, ch. 6, 7.1 to 7.3, 8.1, 8.5 (up to p. 199), 9.3 & 9.5 ] 1. Definition and interpretation a. Difference quotient b. Derivative c. Increasing and decreasing functions d. Limits e. Continuity vs differentiability 2. Rules of Differentiation a. Constant fun ...
slides
... • SVD: Singular values are always positive! • Eigen Analysis: Eigen values can be real or imaginary – Real, positive Eigen values represent stretching of the space along the Eigen vector – Real, negative Eigen values represent stretching and reflection (across origin) of Eigen vector – Complex Eigen ...
... • SVD: Singular values are always positive! • Eigen Analysis: Eigen values can be real or imaginary – Real, positive Eigen values represent stretching of the space along the Eigen vector – Real, negative Eigen values represent stretching and reflection (across origin) of Eigen vector – Complex Eigen ...
Beyond Vectors
... V • Let V and W be vector space. • A linear transformation T: V→W is called an isomorphism if it is one-to-one and onto • Invertible linear transform • W and V are isomorphic. Example 1: U: Mmn Mnm defined by U(A) = AT. Example 2: T: P2 R3 ...
... V • Let V and W be vector space. • A linear transformation T: V→W is called an isomorphism if it is one-to-one and onto • Invertible linear transform • W and V are isomorphic. Example 1: U: Mmn Mnm defined by U(A) = AT. Example 2: T: P2 R3 ...
