What property does f have to have if you want to apply the second derivative test

f must be twice continuously differentiable

What multiplicity does lambda have if it is a simple eigenvalue?

Lambda has the multiplicity of 1

Matrix A is invertible if and only if…

…0 is not an eigenvalue of A

When are two vectors orthogonal to each other?

If their dot product is zero

When is a matrix considered to be orthogonal?

If all columns are unit eigen vectors and are pairwise orthogonal

What are properties of an orthogonal matrix?

It can be seen as a transformation which preserves the angles and lenghts of its intputs

When is a matrix considered to be a rank-one matrix?

If it can be written as a vector product

How many eigenvalues does a nxn matrix A have at most?

A has at most n eigenvalues

Is the zero vector included in the defintion of an eigenspace?

Yes it is, although it is not considered an eigenvector

How is the Jacobian matrix of a function f constructed?

Columns same variable but different function

Rows same function, different variables

What is the particularity of the Jacobian with respect to the gradient of a function f?

It is the same as the Hessian

What properties does a subspace of a vector space have?

It is not equal to the empty set

It is closed under addition

It is closed under scalar multiplication

When is a vector space linearly dependent, when is linearly independent

Dependent if: a vector can be expressed by a linear combination of other vectors

Independent: Otherwise

Is the kernel of a matrix A a linear subspace of the complex numbers?

Yes. Indeed

What are the eigenvalues of a triangular matrix?

The diagonal entries

If lambda is an eigenvalue of A, what is an eigenvalue of A^-1?

1/lambda

What is the determinant of an orthogonal matrix?

It is either -1 or 1

Is any eigenspace a subspace of the complex numbers?

Yes. It is

Last changed10 months ago