5

Question 1

Suppose that A is a square matrix. The number λ is said to be an…

Answer 1

eigenvalue of A if for some non-zero vector x, Ax = λx. Any non-zero vector x for which this equation holds is called an eigenvector.

Question 2

Steps to find eigenvalues and eigenvectors of A

Answer 2

1. The eigenvalues of A are the λ values that satisfy Ax = λx, equivalent to Ax = λIx or (A - λI)x = 0. The eigenvectors of A for eigenvalue λ are any non-zero vector x for which this equation holds. This has a solution other than x=0 when determinant I A - λI I =0, meaning we can solve it to find the eigenvalues of A since eigenvectors x can never be zero vectors.
2. Find and state characteristic polynomial p(λ).
3. Factorise p(λ) to find eigenvalues.
4. To find an eigenvector for λ₁ , solve equation (A-λ₁I )x = 0.
5. Any non-zero multiple of this eigenvector is an eigenvector of A for λ₁.

Question 3

Square matrices A and B are similar if

Answer 3

there is an invertible matrix P such that P^-1AP = B.

Question 4

The matrix A is diagonalisable if it is

Answer 4

similar to a diagonal matrix, i.e. if there is a diagonal matrix D such that P^-1AP=D.

Question 5

A matrix A is diagonalisable if and only if

Answer 5

it has n linearly independent igenvectors.

Question 6

Symmetric: matrices (transpose A^T is equal to itself) are always

Answer 6

diagonalisable.

Question 7

A matrix P is orthogonal if

Answer 7

P^TP = PP^T = I, i.e. if P has inverse P^T.

Question 8

A matrix A is orthogonally diagonalisable if

Answer 8

there is an orthogonal matrix P such that
P^TAP = D where D is a diagonal matrix.

Question 9

The condition that P^TP = I (for matrix P to be orthogonal) means

Answer 9

• any two of the eigenvectors x₁,.., x_n must
  be orthogonal x_i^Tx_j = 0 (i ≠ j)
• each eigenvector must have length 1, i.e. x_i^Tx_j = 0

Question 10

If the matrix A is symmetric (A^T = A) then

Answer 10

• eigenvectors corresponding to different eigenvalues are orthogonal.
• A is orthogonally diagonalisable

Question 11

Length of a vector x is

Answer 11

IIxII = √(Σx_i²) (i=1 to n)

Question 12

When to normalise vectors of P

Answer 12

• When A is a symmetric matrix so P^-1=P^T
• When D = P^TAP
• Find an orthogonal matrix P