Pure Mathematics C4: Eigenvalues & Eigenvectors

26 flashcards · Mathematics · by lakesare

What's another word for eigenvalue?
Characteristic value.
What's v and λ?
v - eigenvector λ - its corresponding eigenvalue
Visually imagine all eigenvectors with their corresponding eigenvalues of the transformation "reflection in x-axis".
λ = 1, eigenvectors = { (k, 0) : k ≠ 0 } λ = –1, eigenvectors = { (0, k) : k ≠ 0 } It is clear geometrically that every other line through the origin is changed by this transformation: we can find no other vector that maps to a multiple of itself. The only eigenvectors of t are therefore of the forms (k, 0) with eigenvalue 1 and (0, k) with eigenvalue −1, where k = 0.
What's eigenvector, in your own words?
Such point that it maps to a scalar multiple of itself.
[Finding eigenvectors&values] Name 3 interesting ways to find eigenvectors.
consider images of various points In the example we found two lines that are mapped to themselves by t, by considering the images of various points. This is a rather hit-and-miss way of finding eigenvalues and eigenvectors.geometric It is sometimes possible to do so by considering the geometry of the transformation.general method - use characteristic equation and eigenvector equations.
[Finding eigenvectors&values] What are the names of these equations?
characteristic equation Helps us find eigenvalues (another name for eigenvalue is "characteristic value").eigenvector equation Helps us find eigenvectors.
[Finding eigenvectors&values] What's an excellent check for whether you found the right eigenvalues?
Tr A = ∑ eigenvalues of A The diagonal entries are 2, 2 and 2, which sum to 6.The eigenvalues are 3, 2 and 1, which also sum to 6.
[Finding eigenvectors&values] Why is it easy to find eigenvalues of these matrices?
Because they are triangular and diagonal - eigenvalues are just entries on the diagonal.
[Eigenspace] What's eigenspace of t corresponding to λ?
Eigenspace of t corresponding to λ = { all eigenvectors corresponding to λ + 0 } Since S(λ) is a subspace comprising eigenvectors (and 0), we call it an eigenspace.
What's S(λ)?
Eigenspace of t corresponding to the eigenvalue λ.
[Finding eigenvectors&values] Given this characteristic equation, what's the multiplicity of each eigenvalue λ? Characteristic equation
λ = 0 has multiplicity 1 λ = 1 has multiplicity 2 The eigenvalue λ = 1 is a ‘repeated’ solution of this characteristic equation; it is a multiple root and we say that λ = 1 has multiplicity 2 because the factor (λ − 1) occurs twice.
[Eigenvector basis] What's eigenvector basis of t? Eigenvectors of t, where t is the reflection across the x-axis.
{ 1 eigenvector for λ1, 1 eigenvector for λ2, etc. }
[Eigenvector basis VS eigenspace] Eigenspace S(λ) VS eigenvector basis? Eigenvectors of t, where t is the reflection across the x-axis.
eigenspace of t = { eigenvectors corresponding to λ + 0 } Since S(λ) is a subspace comprising eigenvectors (and 0), we call it an eigenspace.eigenvector basis of t = { 1 eigenvector for λ1, 1 eigenvector for λ2, etc. }
[Eigenvector basis] What does matrix associated with t look like if we find it with with respect to the { 1 eigenvector for λ1, 1 eigenvector for λn }? eigenvector basis of t = { 1 eigenvector for λ1, 1 eigenvector for λn }
[Eigenvector basis] What do we mean when we say "eigenvector basis of matrix A"?
Just the eigenvector basis for the corresponding transformation t.
[Eigenvector basis] Give an example of an eigenvector basis for A.
Just take 1 eigenvector corresponding to each λ.
[Diagonalisation] When is the matrix A called diagonalisable?
When we can find P such that P−1AP = Diagonal matrix Clearly the matrices A, D and P must all be square matrices of the same size.
[Diagonalisation] Name one use-case for diagonalising a matrix.
To find powers of the matrix (An).
[Diagonalisation] When asked to diagonalise the matrix, what should we return?
Both the diagonalised matrix D, and transition matrix P.
[Transition matrix] What is the transition matrix P supposed to do?
Converts our vector from basis E to standard basis.
[Transition matrix] How would we construct the transition matrix? Transition matrix P converts our vector:- from basis E- to standard basis.
What's strange about the definition of the orthogonal matrix?
Orthogonal matrix - matrix whose columns are orthonormal vectors. It is important to remember that the columns of an orthogonal matrix are orthonormal vectors, not just orthogonal vectors, despite the name!
[Diagonalisation] Diagonalisable matrix VS orthogonally diagonalisable matrix?
Diagonalisable matrix - A = PDP–1 for some P Diagonalisable matrix - A = PDP–1, but P is orthogonal
[Conics and quadrics] Conics VS quadrics?
Quadrics are the 3-dimensional analogues of conics. Quadrics, or quadric surfaces, are surfaces in R3.
[Conics and quadrics] Which quadrics will we consider in this unit? It can be shown that there are nine types of quadrics involving curved surfaces in R3. There are 5 types of non-degenerate quadric: • the ellipsoid (which includes the sphere) • the elliptic paraboloid • the hyperbolic paraboloid • the hyperboloid of one sheet • the hyperboloid of two sheets. There are 4 types of degenerate quadric involving curved surfaces: • the elliptic cone • the elliptic cylinder • the parabolic cylinder •…
6 quadrics (5 non-degenerate and 1 elliptic cone). the ellipsoid (including the sphere), the elliptic paraboloid, the hyperbolic paraboloid, the hyperboloid of one sheet, the hyperboloid of two sheets and the elliptic cone
[Conics and quadrics] Say in your own words - how do we determine the type of a conic/quadric?
Put quadric in standard position: align the axes, translate the origin. We now look at how we can write the equation of a quadric in standard form. We align the axes and translate the origin appropriately by using the same techniques that we used for conics: introducing matrices, orthogonal diagonalisation and completing the square.