Linear Algebra

Invertible Matrix

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix over a continuous uniform distribution on its entries, it will almost surely not be singular.

Matrix A multiplying matrix B (usually a vector) represents a linear function. f(B)=AB.
f(cX)=cf(X), for constant c
This means that in a geometrical sense, origin is fixed. As example, reflection, rotation, scaling, ... can be represented with a linear transformation;
affine transformation: a linear transformation followed by a translation  x -> Ax+b
 Affine transformation can be represented as a linear transformation in a higher dimension
is equal to y=Ax+b but in one dimension higher.

Eigenvector (of a matrix)
If matrix acts on these vectors changes their magnitude, but doesn't change their direction (Just possibly reversing it).
Matrix acts on an eigenvector by multiplying its magnitude by a factor (changes the magnitude, and if the value of the factor is negative, reverses the direction).
This factor is the Eigenvalue associated with that eigenvector.

Set of all eigenvectors that have the same eigenvalue, together with the zero vector.

If matrix A is a linear transformation, a non-null vector x is an eigenvector of A if there is a scalar λ such that
Ax = λx
The scalar λ is said to be an eigenvalue of A corresponding to the eigenvector x.

Benefits of knowing the eigenvectors (and values) of a Matrix

the effects of the action of the matrix on the system can be predicted.

if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used


Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation

A\, \mathbf{x} = \lambda\, \mathbf{x}
for some scalar λ. In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x.

only certain special vectors x are eigenvectors, and only certain special scalars λ are eigenvalues.
If λ = 1, the vector remains unchanged (unaffected by the transformation). Identity transformation
λ = 1, reflection
To get the eigenvectors:
Ax = λx  -->  Ax-λx=0  -->  (A-λI)x=0   --> to avoid only getting the trivial solution (A-λI) should be non-inversible[because it is equal to zero!!!], so the det = 0
Characteristic Polinomial
if  det(A-λI)=0  -->   we have one equation, and only one variable λ, hence λ is easily solved!

Subpages (2): Derivative SVD