Describe orthogonal vectors
2 vectors are orthogonal to each other if they are perpendicular, ie if their dot product = 0
A set of vectors are orthonormal/form an orthonormal basis if all vectors have a magnitude of 1 and if they are all orthogonal to one another
Give the 2x2 matrix that rotates a vector α degrees anti-clockwise
Describe the determinant of a matrix product
For any square matrix, det(AB) = det(A) x det(B)
Give the equation for the cofactor, cij of an element aij
Where Mij (or minor) is the matrix left over when the elements in row i and column j have been removed
Define the Jacobian
|J| (the determinant of the transformation matrix J) is the Jacobian, which gives the factor by which the volume element changes in a transformation.
Describe 6 properties of determinants
1) Swapping rows and columns does not change the determinant
2) A determinant vanishes if one of the rows or columns contains only zeroes
3) If a row or column is multiplied by a constant, the determinant will also be multiplied by that constant.
4) A determinant vanishes if two rows or columns are multiples of each other
5) If we interchange a pair of rows or columns the determinant changes sign
6) Adding a multiple of one row or column to another doesn't change the value of the determinant
Describe the transpose of a matrix, A
AT is obtained by switching the rows for columns. Starting with an m x n matrix, an n x m will be obtained.
Describe the transposition of matrix products
Transposing a product of matrices reverses the order of multiplication, despite how many elements are involved. (ABC)T = CTBTAT
Describe symmetrical matrices
If AT = A, A is symmetric.
If AT = -1, A is antisymmetric.
Describe the product of orthogonal matrices
If A and B are orthogonal matrices, their product, C = AB will also be an orthogonal matrix
Describe how to take the complex conjugate of a matrix
Take the complex conjugate of each element. (A*)ij = a*ij
Note: if A = A*, the matrix is real
Describe hermitian conjugation
Also known as the Hermitian adjoint and represented by a superscript dagger (†), it is a combination of complex conjugation and transposition, in either order: A† = (AT)* = (A*)T
If A† = A, A is Hermitian
If A† = -A, A is anti-Hermitian
Describe how to work out the Hermitian conjugate of a matrix product
(AB)† = B†A†
Describe the trace of a matrix
The trace of a matrix is the sum of all diagonal elements.
Tr(A) = a11 + a22 + a33 + ....
Describe how to work out the determinant of a matrix A
Multiply each element of one row or column by its cofactor and add the results. Eg for a 3x3 matrix, the determinant can be calculated by working out a11c11 + a12c12 + a13c13
Describe the inverse of a matrix A
The inverse of a matrix, defined as A-1 is given such that AA-1 = I
C is the matrix made up of the cofactors of A. CT is known as the adjoint matrix to A, denoted by Aadj
Describe the Kronecker delta, δij
Describe a unitary matrix
A square matrix U is unitary if U†U = UU† = I
Describe linearly dependent vectors
A set of vectors X1, X2,..., Xn are linearly dependent if constants ci cant be found (not all zeroes) such that c1X1 + c2X2 + ... + cnXn = 0. If no such constants exist, then Xi are linearly independent.
Give the equation of a diagonalised matrix
D = L-1ML
M is the diagonalisable matrix
L is the matrix whose columns are the eigenvectors of M
Give the equation for the diagonalisable matrix M to any power
Mn = LDnL-1
Describe a normal matrix
A matrix is described as normal if M†M = MM†
Give the equation for the inverse of a matrix
Where C is the matrix made up of the cofactors of A. C is knowns as the adjoint matrix to A, Aadj
State a property of all normal matrices
All normal matrices are all diagonalisable. Unitary, orthogonal, hermitian, and real symmetric matrices are all normal and hence diagonalisable
Describe a hermitian matrix
A square matrix H is said to be hermitian if H† = H
Describe a real and symmetric matrix
A matrix is real and symmetric if its entries are real and ST = S
Describe an orthogonal matrix
A square matrix O is orthogonal if it has real entries and OTO = OOT = I
Give the compact equation for finding the factor by which the volume element changes when we make a transformation.
dr' = Jdr
Describe how to test for linear dependence using determinants
The column vectors in question can be written as a matrix - columns forming columns. The determinant of this matrix M is taken.
If det(M) = 0, the vectors are linearly depedent
If det(M) ≠ 0, the vectors are linearly independent
Give the equation needed to work out eigenvectors
Mv = λv
Give the determinant of a 2x2 matrix A
DetA = a11a22-a12a21
Give the characteristic equation used to find eigenvalues
Define a diagonal matrix
A square matrix with elements only along the diagonal
Describe Cramer's rule