8- Eigen Vectors, Projections and Inner Product

Question 1

What is an eigenvector of a matrix Anxn?
In particular what does it represent?

What is an eigenvalue?
What does it represent?

Answer 1

• It is a non zero vector with n entries x such that Ax=lambda*x for some scalar lambda (eigenvalue).
• It is a vector x that does not change direction when A is applied to it.
• lambda such that Ax= lambda*x
• how much the eigenvector is stretched or compressed when A is applied to it.

Question 2

From Ax = lambda*x, if we are given the eigenvalues how can we find the eigenvectors?
how can we find the eigenvalues?

Answer 2

Ax = lambdax;
Ax - lambdaIx= 0;
(A-lambdaI)x = 0;
-> we look for the solutions.

det(A-lambda*I) = 0

Question 3

What is the eigenbasis?

What is the eigenspace?

Answer 3

• set of eigenvectors of a matrix
• set of all the eigenvectors associated with a particular eigenvalue lambda: given A and lambda, the set of vectors v such that Av = lambdav -> (A-lambdaI)x=0 -> Nul(A-lambda*I).

Question 4

What is a triangular matrix?

What are the eigenvalues of a triangular matrix?

What is the determinant of a matrix?
Is a matrix invertible if it has a zero eigenvalue?

Answer 4

• a square matrix where all entries are zero either above or below the main diagonal.
• the eigenvalues of a triangular matrix are the entries in the main diagonal.
• the determinant of a matrix is equal to the product of its eigenvalues.
• If one eigenvalue is 0 then the determinant is 0 and so the matrix is not invertible.

Question 5

If we perform row operations on a matrix A, its determinant changes…how?

Answer 5

• replacement: does not modify the determinant.
• interchange: detA= -1 * detB
• scaling: detA= 1/c * detB

Question 6

Diagonal Matrix:
- when can we say that a matrix A is diagonalizable?
- why do we need it? thanks to which math. property of matrices?
- when is Anxn diagonalizable? How can we diagonalize it?

Answer 6

• if A is similar to a diagonal matrix that is A= PDP^-1 for some invertible matrix P and some diagonal matrix D.
• It allows us to quickly calculate A^k = P(A^k)P^-1. Thanks to the associativity of matrices.
• It has n linearly independent eigenvectors.
  The diagonal entries of D are the eigenvalues of A that correspond to the eigenvectors that are located in P.

Question 7

What is a dot product between two vectors?
What is the L2 distance, how is it also called?
Given a scalar c and a vector v, what is the L2 of c*v?

• If 2 vectors v and w are orthogonal than it means that…

Answer 7

• math operation that takes to vectors as an input and returns a scalar.
• Also called norm of a vector, it is the sqrt of the sum of the squares of the entries of the vector.
• that L2((v+w)^2) = L2(v^2) + L2(w^2)

Question 8

When can we say that 2 vectors v and w are orthogonal? What does it mean?

Answer 8

• if their dot product is zero. It means that each vector has no component in the direction of the other.

Question 9

Which the vector that represents the projection of vector v onto vector p? What does it represent?

Answer 9

• it is v^ = (v*p)/(p^p) * p
• It represents how much v points toward p.

Question 10

How to find the vector that represents the distance from vector v to vector p?

Answer 10

• I find the projection of v onto p;
• I find the orthogonal vector = v - v^
• I calculate the norm of the orthogonal vector.aaA