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2. Eigenvalues and eigenvectors - Wikipedia

   en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

   Principal component analysis of the correlation matrix provides an orthogonal basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.

3. Singular value - Wikipedia

   en.wikipedia.org/wiki/Singular_value

   Schmidt called singular values "eigenvalues" at that time. The name "singular value" was first quoted by Smithies in 1937. In 1957, Allahverdiev proved the following characterization of the nth singular number: [5]

4. Rank (linear algebra) - Wikipedia

   en.wikipedia.org/wiki/Rank_(linear_algebra)

   This number (i.e., the number of linearly independent rows or columns) is simply called the rank of A. A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank-deficient if it does not

5. Eigenvalue algorithm - Wikipedia

   en.wikipedia.org/wiki/Eigenvalue_algorithm

   Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...

6. Eigendecomposition of a matrix - Wikipedia

   en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

   Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.

7. Rank–nullity theorem - Wikipedia

   en.wikipedia.org/wiki/Rank–nullity_theorem

   The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M ; and the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f ) and the nullity of f (the dimension of the kernel of f ).