Notes on k-wedge vectors, determinants, and characteristic
... Although the Cayley–Hamilton theorem does hold over arbitrary fields (including weird ones like F2 ), we will only prove it for real and complex operators. For complex operators this will be easy, once we know that every complex operator is upper-triangular w.r.t. some basis. Proposition 4.5. If T ∈ ...
... Although the Cayley–Hamilton theorem does hold over arbitrary fields (including weird ones like F2 ), we will only prove it for real and complex operators. For complex operators this will be easy, once we know that every complex operator is upper-triangular w.r.t. some basis. Proposition 4.5. If T ∈ ...
CHAP03 Vectors and Matrices in 3 Dimensions
... Most of the theory of 2-dimensional vectors can be extended in a straightforward way to 3dimensional vectors. In most cases the only change that needs to be made is to change “2” into “3” x2 x1 ...
... Most of the theory of 2-dimensional vectors can be extended in a straightforward way to 3dimensional vectors. In most cases the only change that needs to be made is to change “2” into “3” x2 x1 ...
The Fundamental Theorem of Linear Algebra Gilbert Strang The
... side-the splitting of b into p + e. ...
... side-the splitting of b into p + e. ...
Partial Derivatives
... If we want to find the maximum and minimum value of function f under condition(s) or constraint(s), then we can apply the method of Lagrange multipliers. Suppose we want to minimize the function f (x, y) under the constraint, function g (x, y) = k, then we define the Lagrange function as the followi ...
... If we want to find the maximum and minimum value of function f under condition(s) or constraint(s), then we can apply the method of Lagrange multipliers. Suppose we want to minimize the function f (x, y) under the constraint, function g (x, y) = k, then we define the Lagrange function as the followi ...
Revised Version 090907
... complex number, z = x + yi , can be represented uniquely by a point with coordinates ( x, y ) on the complex plane. Multiplying a complex number by -1 can be represented as the rotation of a point € plane 180º counterclockwise about the origin to another point on on the complex the€complex plane equ ...
... complex number, z = x + yi , can be represented uniquely by a point with coordinates ( x, y ) on the complex plane. Multiplying a complex number by -1 can be represented as the rotation of a point € plane 180º counterclockwise about the origin to another point on on the complex the€complex plane equ ...
Matrix manipulations
... Data sparsity and fast matrix-vector products We say a matrix A ∈ Rn×m is data sparse if we can represent it with far fewer than nm parameters. For example, • Sparse matrices are data sparse – we only need to explicitly know the positions and values of the nonzero elements. • A rank one matrix is da ...
... Data sparsity and fast matrix-vector products We say a matrix A ∈ Rn×m is data sparse if we can represent it with far fewer than nm parameters. For example, • Sparse matrices are data sparse – we only need to explicitly know the positions and values of the nonzero elements. • A rank one matrix is da ...
