Matrix Decomposition

Synonyms

Matrix factorization

Notations and Glossary

Notations

ℝ:

The set of all real numbers

ℂ:

The set of all complex numbers $\left\{a+b\mathbf{i}:a,b\in \mathbb{R},\mathbf{i}=\sqrt{-1}\right\}$

ℝⁿ:

The set of all real column n-vectors

ℂⁿ:

The set of all complex column n-vectors

ℝ^m×n:

The set of all m × n real matrices

ℂ^m×n:

The set of all m × n complex matrices

I_k:

The identity matrix of order k

O:

The zero matrix of appropriate order

A^T:

The transpose of matrix A

A*:

The conjugate transpose of matrix A

A⁻¹:

The inverse of matrix A

||x||:

The Euclidean norm of x ∈ ℂⁿ defined by $\mid {\left|\mathbf{x}\right|}^2=\sum\limits_{i=1}^n{\left|{x}_i\right|}^2$

||A||_F:

The Frobenius norm of A = [a_ij] ∈ ℂ^m×n defined by $\mid {\left|A\right|}_F^2=\sum\limits_{i=1}^m\sum\limits_{j=1}^n{\left|{a}_{ij}\right|}^2$

tr (A):

The trace of A ∈ ℂ^n×n given by $\mathrm{tr}(A)=\sum\limits_{i=1}^n{a}_{ii}$

A ⊕ B:

The direct sum of matrices A and B

diag (A₁,…, A_k):

The block diagonal matrix...