Further Linear Algebra Flashcards
(17 cards)
Orthogonal matrix
Preserves the value of the inner product
Complex conjugate
Matrix obtained by taking the complex conjugate of each of the elements of A
Conjugate transpose
Transpose of the complex conjugate
Dagger
The transpose of the complex conjugate
Hermitian
If the original matrix is equal to its dagger. Elements on main diagonal must be real. (A real one is a symmetric matrix) eigenvalues are real
Skew hermitian
If the dagger of the matrix is equal to the negative of the original matrix. The diagonal must be pure imaginary or 0 (real one is an antisymmetric matrix) eigenvalues are pure imaginary or zero
Unitary
If the dagger is equal to the inverse of the matrix (real one is an orthogonal matrix). The eigenvalues have absolute value of 1
Inner product
Provides a measure of the relationship between two vectors (dot product)
Unitary transformation
Preserves value of the inner product, hence also the lentth of a complex vector
A hermitian, skew hermitian or unitary matrix has…
A basis eigenvectors for C^n that is a unitary system
Similar matrices
Matrices with same determinant and eigenvalues
Similarity transformation
If a transformation matrix s is unitary then an orthonormal basis will transform into an orthonormal basis
Quadratic form
Any polynomial function of the elements of an n-dimensional column vector in which every term is of degree 2
Standard form of the quadratic form
When there are no cross terms (eg x1x2) in the quadratic form
Positive definite
All eigenvalues are positive and greater than 0
Energy characteristic equation
|B - ω^2A| where B is the matrix found from the potential energy and A is the matrix found from the kinetic energy((should be identity matrix)
Substitution for lambda
mω^2 /k = λ
