Matrices Revision Flashcards
The Laplace expansion
det(A) = NΣ i=1 AijCij
where Cij = (-1)^(i+j)Mij
where Mij is the determinant of the minor
The determinant of a general diagonal matrix is
the product (the multiples) of the diagonals.
Volume of a parallelepiped is
V = a.(b x c)
Inverse of a matrix =
A^(-1) = C^(T)/det(A)
where T is the transpose
and C is the cofactor matrix i.e. the matrix of values Cij = (-1)^(i+j)Mij
how to find Eigenvalues
det(A-λI)=0
how to find eigenvectors
substitute eigenvalues into det(A-λI)
then solve the new matrix for
det(A-λI) x(vector) = 0
where x is the vector of (x1,x2,x3)
how to take the normal
1/(sqrt(x1^2+x2^2+x3^2))
Generally the eigenvalues of a diagonal matrix
are the matrix elements along the diagonal
a matrix A is hermitian if
it is the same as its Hermitian conjugate
A† =
(A^(T))^* = A
derivative of a log is
log(x)’ = 1/x
or log(nx) = 1/nx *(nx)’
x, y and z in polar coordinates
x = rcosθ
y = rsinθ
z = rcosθsinϕ
r^2 =
x^2 + y^2
x, y and z in spherical coordinates
x = rsinθcosϕ
y = rsinθsinϕ
z = rcosθ
