Week 14 Flashcards
(18 cards)
Algebraic Multiplicity def ?
the largest integer k such
that (x -λ)k is a factor of the characteristic polynomial l A- λI l
Geometric multiplicity?
The dimension of the eigenspace relative to the eigenvalue
Invertable matrix?
AA^-1 = I
When is A diagonalisable
for A = PDP^-1
Sum of A.M = n (FOR REAL ROOTS)
For each eigenvalue the A.M (λ) = the G.M (λ)
Orthonormal basis for basis of diagonalisable matrix?
Just normal as basis P is already linearly independent e.g. orthogonal
Only if we have more than 1 vector in any eigenspace then we use Gran Schmidt, if not we just normal
Quadratic form?
q(x) = x^T A x
where A=A^T
How to construct quadratic form from cartesian?
The square terms are the diagonal terms in A, then the rest is split then go around in ORDER
What is positive definite , semi positive definite , negative definite and semi negative definite using quadratic form?
positive definite if q(x)≥ 0 and q(x)=0 only if x=0
semi positive definite if q(x)≥0 for all x
negative definite if q(x)≤0 and q(x)=0 only if x=0
semi negative definite if q(x)≤0
if none of these then INDEFINITE
USE EIGENVALUES OF A TO CALC
What do eigenvalues tell us about q(x) and A (quadratic form)?
Positive definite all eigenvalues are positive
Semi Positive definite all eigenvalues are non negative
Negative definite all eigenvalues are negative
Semi negative definite all eigenvalues are non positive
Indefinite 1 positive and 1 negative eigenvalue
For null space what are fuck ups?
If someting = 0 then we just leave it out as equals 0
if 0 1 0
0 0 1
0 0 0 then = (1 , 0, 0) the vector
Eigenvalues distinct?
Eigenvectors are linearly independent and form a basis
Definition of a square matrix be diagonalisable?
if there exists an invertible matrix P and a diagonal matrix D such that P^1 A P=D
Orthogonal diagonalise?
find orthonormal basis
here P^-1 = P^T
Drawing graph key eq to shift axis to make easier?
(V) = PB (V)B
where (V)B = (X Y)
then eq here can change original cartesian to easier thing on new axis (normally ACW rot)
Hyperbola eq?
Horizontal hyperbola = x^2/a^2 - y^2/a^2 = 1
asympnotes are y=+- b/a x
Vertical hyperbola= y^2/b^2 - x^2/a^2 = 1
asympnotes are y=+- b/a x
When calc eigenspaces where to put negative?
keep negative at the bottom
When using transition matrix to help draw graph, when we get final eq what do we remove?
the xy term
Comment when removing xy term?
Substituting may still leave a cross term if the axes aren’t fully aligned; diagonalisation removes it by rotating the system to the conic’s principal axes
