Maths Flashcards
(29 cards)
Greens function
Solve homogeneous, initial conditions, continuity and jump,
Similarity transform
The expression of a matrix in a different basis
Eigenvector and eigenvalue
(A-lambda E) x =0. |A- lambda E|=0
Commute
AB-BA=0
Outer product
Produces a matrix. Each term in a vector multiplied by one of the terms in the other vector
Symmetric matrix
Matrix is equal to its transpose.
Hermitian
Matrix is equal to the complex conjugate of the transpose
Orthogonal
The inverse it equal to the transpose
Closure
Performing an operation on two of the elements in the group and producing another element in the group
Associativity
The order of applying the operation doesn’t matter
Identify
There is an element in the group which is applied to be another element using the operation with no change to the element.
Inevitability
When the operation is applied between two elements such that the zero matrix is formed
Commutative
f(a,b)=f(b,a)
What makes a field
(A,+) is an abelian group with a neutral element O, (A{O},*) is an abelian group with a neutral element
Continuity
The two terms in the greens function are equal at x_0
Jump
The derivative of the positive component at x_0 minus the derivative of the negative component at x_0 is equal to one
Vector space
f(x+y)=f(x)+f(y), f(ax)=af(x) these say there are linear maps on the space. Vector space is a group plus an external product. (a+b)x=ax+bx
Change of basis
Find eigenvalues and eigenvectors. The eigen vector go vertically into a matrix S. B=S^-1AS
Diagonalizing a matrix
Solving eignevalues and putting them in the diagonal
Exponential and matrix to the power
If diagonal then just perform the actions on the terms
Unitary
UU^H=I
Relate cos and sin expansions to sinh and cosh
Rather that plus and minus they are both all plus
Cauchy-Riemann condition
f(z)=f(x+iy)=u(x,y)+iv(x,y). The condition is du/dx=dv/dy and dv/dx=-du/dy
Taylor series
Sum for zero to infinity of f^n(x_0)/n! (x-x_0)^n
