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 changeda year ago