$$ <a,b>= \sum_{i=1}^D a_i * b_i $$
$$ ||a||2 = \sqrt{<a, a>} = \sqrt{\sum{i=1}^D a_i^2} $$
Number of linearly independent columns and rows in the matrix.
Its equal to the number of non-zero singular values.
For a matrix A the inverse is such a matrix that multiplied by A from either side produces an identity matrix.
$$ A A^{-1} = I, A^{-1} A = I $$
It exists if A is square and the determinant is non-zero.
An eigenvector x of matrix A is such a vector that remains in the same direction after being transformed by a linear operator A.
The constant lambda is said to be the corresponding eigenvalue.
$$ A x = \lambda x $$
$$ X w = y \\ X^T X w = X^T y \\ w = (X^T X)^{-1} X ^ T y $$
SVD is a way to factorize a matrix.