Px153 Revision Flashcards
(27 cards)
How can you tell if a matrix has an inverse?
If the determinant is non zero it has an inverse
What are the requirements for convergence in the mean to occur?
Only that integral (f(x))² dx over the interval is less than infinity
When does piecewise convergence occur?
When the function is continuous over the interval except for a finite number of points, essentially there are breaks in the function
When does uniform convergence occur?
When the function is continuous over the interval and at periodic extensions
What matrices can be expressed in LU decompositions?
All square, non singular matrices
What is a hermitian matrix?
One where the matrix is equal to it’s conjugate transpose
What is a unitary matrix?
One where the inverse is equal to it’s hermitian (conjugate transpose) or the matrix multiplied by it’s hermitian is equal to the identity matrix
What is an orthogonal matrix?
One where it’s transpose is equal to it’s inverse
What is an antisymmetric matrix?
One where the negative is equal to the transpose of the original matrix
What is the definition of a conservative field?
A function for which the line integral vanishes for all possible closed paths, and can be expressed as the gradient of a potential function
What is the difference between the sine and cosine series of a function?
The sine series is antisymmetric, the cosine series is symmetric
How do you work out the inverse of a matrix?
Adjugate matrix/determinant
How to work out the adjugate of a matrix?
Work out the matrix of cofactors then transpose it
What is the cofactor of a matrix element?
The determinant of the minor matrix corresponding to that element
What is the commutator of the matrices A and B?
AB - BA = Commutator
When are eigenvectors orthogonal (I think?)
When the matrix is symmetric or hermitian
Which Fourier series coefficients do you need to worry about if the function is symmetric?
Only Bn
What Fourier series coefficients do you need to worry about if the function is antisymmetric?
A0 and An
What is the formula for A0?
= 1/L integral between L, -L( F(x) dx)
What is the formula for An in Fourier series?
= 1/L integral between L, -L( F(x) cos(n pi x/ L) dx)
What is the formula for Bn in Fourier series?
= 1/L integral between L, -L( F(x) sin(n pi x/ L) dx)
When do two equations have infinite solutions?
When two equations are linearly dependent (the same within a constant) or if one line is 0=0
What is the definition of a conservative field?
One where the work done around any closed path is 0, and can be expressed as the gradient of a potential
How to get from a coordinate in one basis to a coordinate in another basis?
Multiply the original coordinates by the inverse transformation matrix
