|
Size: 1348
Comment: Simplifying matrix page names
|
← Revision 13 as of 2025-09-24 17:59:57 ⇥
Size: 0
Comment: Simplifying matrix page names
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
| = Matrix Properties = Matrices can be categorized by whether or not they feature certain '''properties'''. <<TableOfContents>> ---- == Idempotency == An '''idempotent''' matrix can be multiplied by some matrix '''''A''''' any number of times and the first product will continue to be returned. In other words, '''''A'''^2^ = '''A'''''. For example, the [[LinearAlgebra/Projections|projection matrix]] '''''P''''' is characterized as '''H'''('''H'''^T^'''H''')^-1^'''H'''^T^. If this were squared to '''H'''('''H'''^T^'''H''')^-1^'''H'''^T^'''H'''('''H'''^T^'''H''')^-1^'''H'''^T^, then per the core principle of [[LinearAlgebra/MatrixInversion|inversion]] (i.e., '''''AA'''^-1^ = '''I'''''), half of the terms would cancel out. '''''P'''^2^ = '''P'''''. Only a square matrix can be idempotent. ---- == Positive Definite == A '''positive definite matrix''' is a symmetric matrix where all [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]] are positive. Following from the properties of all symmetric matrices, all pivots are also positive. Necessarily the [[LinearAlgebra/Determinants|determinant]] is also positive, and all subdeterminants are also positive. === Positive Semi-definite === A slight modification of the above requirement: 0 is also allowable. ---- CategoryRicottone |
