Matrix analysis

Matrix analysis is a subfield of linear algebra. It focuses on analytical properties of matrices. In this subject, vector norms and matrix norms are introduced. The goal of this area is deepen understanding to matrix eigenvalues and system of linear equations. This leads to discussions in numerical linear algebra.^{[1][2][3][4][5]}

Main Topics

The following topics are studied in the context of matrix analysis:^{[1][2][3][4][5]}

• Discussing matrices by tools from functional analysis
• Inequalities related to matrix norms or matrix eigenvalues^[6][7][8][9]
• Behavior of matrix eigenvalues

Significance

Functional analysis usually discusses mathematical operators in infinite dimension Hilbert spaces.^[10] But difficulty remains even discussion is limited to matrices (which is a finite dimension mathematical operator). This is because difficulty comes not only from infinite dimension but also non-commutativity.^[11][12][13] And matrices are good examples of non-commutative mathematical operators (In other words, you cannot change the order of matrix multiplication). Matrix analysis is trying to overcome problems caused by non-commutativity.^{[1][2][3][4][5]}

Achievements

The following results are known as remarkable achievements in this area:

• Matrix Young inequalities^[14]
• Lidskii theorem^[4][15]
• Hansen-Pedersen method^[4][16][17]

Journals

The following journals include articles about matrix analysis:

• SIAM Journal on Matrix Analysis and Applications (Published by the Society for Industrial and Applied Mathematics)
• Linear Algebra and its Applications
• Linear and Multilinear Algebra
• The Electronic Journal of Linear Algebra (Published by the International Linear Algebra Society)