HomeHome Metamath Proof Explorer
Theorem List (p. 203 of 424)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-27759)
  Hilbert Space Explorer  Hilbert Space Explorer
(27760-29284)
  Users' Mathboxes  Users' Mathboxes
(29285-42322)
 

Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmamucl 20201 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵𝑚 (𝑀 × 𝑃)))
 
Theoremmamuass 20202 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑂 × 𝑃)))    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   𝐺 = (𝑅 maMul ⟨𝑀, 𝑂, 𝑃⟩)    &   𝐻 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐼 = (𝑅 maMul ⟨𝑁, 𝑂, 𝑃⟩)       (𝜑 → ((𝑋𝐹𝑌)𝐺𝑍) = (𝑋𝐻(𝑌𝐼𝑍)))
 
Theoremmamudi 20203 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((𝑋𝑓 + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘𝑓 + (𝑌𝐹𝑍)))
 
Theoremmamudir 20204 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(𝑌𝑓 + 𝑍)) = ((𝑋𝐹𝑌) ∘𝑓 + (𝑋𝐹𝑍)))
 
Theoremmamuvs1 20205 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    · = (.r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘𝑓 · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘𝑓 · (𝑌𝐹𝑍)))
 
Theoremmamuvs2 20206 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
(𝜑𝑅 ∈ CRing)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘𝑓 · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘𝑓 · (𝑋𝐹𝑍)))
 
11.2.2  Square matrices

In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that (𝑁 Mat 𝑅) is a left module, see matlmod 20229. That (𝑁 Mat 𝑅) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (𝑁 Mat 𝑅) is called "matrix ring" or "matrix algebra" already in this subsection.

 
Syntaxcmat 20207 Syntax for the square matrix algebra.
class Mat
 
Definitiondf-mat 20208* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
 
Theoremmatbas0pc 20209 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
 
Theoremmatbas0 20210 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
 
Theoremmatval 20211 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
 
Theoremmatrcl 20212 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑋𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
 
Theoremmatbas 20213 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Base‘𝐺) = (Base‘𝐴))
 
Theoremmatplusg 20214 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (+g𝐺) = (+g𝐴))
 
Theoremmatsca 20215 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
 
Theoremmatvsca 20216 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ( ·𝑠𝐺) = ( ·𝑠𝐴))
 
Theoremmat0 20217 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (0g𝐺) = (0g𝐴))
 
Theoremmatinvg 20218 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (invg𝐺) = (invg𝐴))
 
Theoremmat0op 20219* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁0 ))
 
Theoremmatsca2 20220 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 = (Scalar‘𝐴))
 
Theoremmatbas2 20221 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))
 
Theoremmatbas2i 20222 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 ∈ (𝐾𝑚 (𝑁 × 𝑁)))
 
Theoremmatbas2d 20223* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅𝑉)    &   ((𝜑𝑥𝑁𝑦𝑁) → 𝐶𝐾)       (𝜑 → (𝑥𝑁, 𝑦𝑁𝐶) ∈ 𝐵)
 
Theoremeqmat 20224* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗)))
 
Theoremmatecl 20225 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐼𝑁𝐽𝑁𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)
 
Theoremmatecld 20226 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝑀𝐵)       (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾)
 
Theoremmatplusg2 20227 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑓 + 𝑌))
 
Theoremmatvsca2 20228 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)    &   𝐶 = (𝑁 × 𝑁)       ((𝑋𝐾𝑌𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘𝑓 × 𝑌))
 
Theoremmatlmod 20229 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
 
Theoremmatgrp 20230 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)
 
Theoremmatvscl 20231 Closure of the scalar multiplication in the matrix ring. (lmodvscl 18874 analog.) (Contributed by AV, 27-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶𝐾𝑋𝐵)) → (𝐶 · 𝑋) ∈ 𝐵)
 
Theoremmatsubg 20232 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (-g𝐺) = (-g𝐴))
 
Theoremmatplusgcell 20233 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       (((𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽)))
 
Theoremmatsubgcell 20234 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (-g𝐴)    &    = (-g𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) (𝐼𝑌𝐽)))
 
Theoremmatinvgcell 20235 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = (invg𝑅)    &   𝑊 = (invg𝐴)       ((𝑅 ∈ Ring ∧ 𝑋𝐵 ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑊𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽)))
 
Theoremmatvscacell 20236 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))
 
Theoremmatgsum 20237* Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑖𝑁, 𝑗𝑁𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈)) finSupp 0 )       (𝜑 → (𝐴 Σg (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑅 Σg (𝑦𝐽𝑈))))
 
11.2.3  The matrix algebra

The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 20243) and an associative algebra (see matassa 20244). Additionally, theorems for the identity matrix and transposed matrices are provided.

 
Theoremmatmulr 20238 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → · = (.r𝐴))
 
Theoremmamumat1cl 20239* The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       (𝜑𝐼 ∈ (𝐵𝑚 (𝑀 × 𝑀)))
 
Theoremmat1comp 20240* The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
 
Theoremmamulid 20241* The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑀, 𝑁⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))       (𝜑 → (𝐼𝐹𝑋) = 𝑋)
 
Theoremmamurid 20242* The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑁, 𝑀, 𝑀⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑁 × 𝑀)))       (𝜑 → (𝑋𝐹𝐼) = 𝑋)
 
Theoremmatring 20243 Existence of the matrix ring, see also the statement in [Lang] p. 504: "For a given integer n > 0 the set of square n x n matrices form a ring." (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
 
Theoremmatassa 20244 Existence of the matrix algebra, see also the statement in [Lang] p. 505:"Then Matn(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg)
 
Theoremmatmulcell 20245* Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝑅)    &    × = (.r𝐴)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗)(.r𝑅)(𝑗𝑌𝐽)))))
 
Theoremmpt2matmul 20246* Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝐴)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑁 ∈ Fin)    &   𝑋 = (𝑖𝑁, 𝑗𝑁𝐶)    &   𝑌 = (𝑖𝑁, 𝑗𝑁𝐸)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐶𝐵)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐸𝐵)    &   ((𝜑 ∧ (𝑘 = 𝑖𝑚 = 𝑗)) → 𝐷 = 𝐶)    &   ((𝜑 ∧ (𝑚 = 𝑖𝑙 = 𝑗)) → 𝐹 = 𝐸)    &   ((𝜑𝑘𝑁𝑚𝑁) → 𝐷𝑈)    &   ((𝜑𝑚𝑁𝑙𝑁) → 𝐹𝑊)       (𝜑 → (𝑋 × 𝑌) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑅 Σg (𝑚𝑁 ↦ (𝐷 · 𝐹)))))
 
Theoremmat1 20247* Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))
 
Theoremmat1ov 20248 Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   𝑈 = (1r𝐴)       (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 ))
 
Theoremmat1bas 20249 The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r‘(𝑁 Mat 𝑅))       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1𝐵)
 
Theoremmatsc 20250* The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿𝐾) → (𝐿 · (1r𝐴)) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 )))
 
Theoremofco2 20251 Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)
(((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
 
Theoremoftpos 20252 The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.)
((𝐹𝑉𝐺𝑊) → tpos (𝐹𝑓 𝑅𝐺) = (tpos 𝐹𝑓 𝑅tpos 𝐺))
 
Theoremmattposcl 20253 The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → tpos 𝑀𝐵)
 
Theoremmattpostpos 20254 The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → tpos tpos 𝑀 = 𝑀)
 
Theoremmattposvs 20255 The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)       ((𝑋𝐾𝑌𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌))
 
Theoremmattpos1 20256 The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 )
 
Theoremtposmap 20257 The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
(𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵𝑚 (𝐽 × 𝐼)))
 
Theoremmamutpos 20258 Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐺 = (𝑅 maMul ⟨𝑃, 𝑁, 𝑀⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → tpos (𝑋𝐹𝑌) = (tpos 𝑌𝐺tpos 𝑋))
 
Theoremmattposm 20259 Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝐴)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋))
 
Theoremmatgsumcl 20260* Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑈 Σg (𝑟𝑁 ↦ (𝑟𝑀𝑟))) ∈ (Base‘𝑅))
 
Theoremmadetsumid 20261* The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑃 = ( I ↾ 𝑁)) → (((𝑌𝑆)‘𝑃) · (𝑈 Σg (𝑟𝑁 ↦ ((𝑃𝑟)𝑀𝑟)))) = (𝑈 Σg (𝑟𝑁 ↦ (𝑟𝑀𝑟))))
 
Theoremmatepmcl 20262* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))       ((𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵) → ∀𝑛𝑁 ((𝑄𝑛)𝑀𝑛) ∈ (Base‘𝑅))
 
Theoremmatepm2cl 20263* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))       ((𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵) → ∀𝑛𝑁 (𝑛𝑀(𝑄𝑛)) ∈ (Base‘𝑅))
 
Theoremmadetsmelbas 20264* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑌 = (ℤRHom‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑄𝑃) → (((𝑌𝑆)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))
 
Theoremmadetsmelbas2 20265* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑌 = (ℤRHom‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑄𝑃) → (((𝑌𝑆)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛𝑀(𝑄𝑛))))) ∈ (Base‘𝑅))
 
11.2.4  Matrices of dimension 0 and 1

As already mentioned before, and shown in mat0dimbas0 20266, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). In the following, some properties of the empty matrix are shown, especially that the empty matrix over an arbitrary ring forms a commutative ring, see mat0dimcrng 20270.

For the one-dimensional case, it can be shown that a ring of matrices with dimension 1 is isomorphic to the underlying ring, see mat1ric 20287.

 
Theoremmat0dimbas0 20266 The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.)
(𝑅𝑉 → (Base‘(∅ Mat 𝑅)) = {∅})
 
Theoremmat0dim0 20267 The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (0g𝐴) = ∅)
 
Theoremmat0dimid 20268 The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (1r𝐴) = ∅)
 
Theoremmat0dimscm 20269 The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑋( ·𝑠𝐴)∅) = ∅)
 
Theoremmat0dimcrng 20270 The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → 𝐴 ∈ CRing)
 
Theoremmat1dimelbas 20271* A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (𝑀 ∈ (Base‘𝐴) ↔ ∃𝑟𝐵 𝑀 = {⟨𝑂, 𝑟⟩}))
 
Theoremmat1dimbas 20272 A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐵) → {⟨𝑂, 𝑋⟩} ∈ (Base‘𝐴))
 
Theoremmat1dim0 20273 The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (0g𝐴) = {⟨𝑂, (0g𝑅)⟩})
 
Theoremmat1dimid 20274 The identity of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (1r𝐴) = {⟨𝑂, (1r𝑅)⟩})
 
Theoremmat1dimscm 20275 The scalar multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       (((𝑅 ∈ Ring ∧ 𝐸𝑉) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( ·𝑠𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r𝑅)𝑌)⟩})
 
Theoremmat1dimmul 20276 The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) (Proof shortened by AV, 18-Apr-2021.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       (((𝑅 ∈ Ring ∧ 𝐸𝑉) ∧ (𝑋𝐵𝑌𝐵)) → ({⟨𝑂, 𝑋⟩} (.r𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r𝑅)𝑌)⟩})
 
Theoremmat1dimcrng 20277 The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ CRing ∧ 𝐸𝑉) → 𝐴 ∈ CRing)
 
Theoremmat1f1o 20278* There is a 1-1 function from a ring onto the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹:𝐾1-1-onto𝐵)
 
Theoremmat1rhmval 20279* The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})
 
Theoremmat1rhmelval 20280* The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)
 
Theoremmat1rhmcl 20281* The value of the ring homomorphism 𝐹 is a matrix with dimension 1. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) ∈ 𝐵)
 
Theoremmat1f 20282* There is a function from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹:𝐾𝐵)
 
Theoremmat1ghm 20283* There is a group homomorphism from the additive group of a ring to the additive group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 GrpHom 𝐴))
 
Theoremmat1mhm 20284* There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})    &   𝑀 = (mulGrp‘𝑅)    &   𝑁 = (mulGrp‘𝐴)       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑀 MndHom 𝑁))
 
Theoremmat1rhm 20285* There is a ring homomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 RingHom 𝐴))
 
Theoremmat1rngiso 20286* There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑂 = ⟨𝐸, 𝐸    &   𝐹 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 RingIso 𝐴))
 
Theoremmat1ric 20287 A ring is isomorphic to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 30-Dec-2019.)
𝐴 = ({𝐸} Mat 𝑅)       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝑅𝑟 𝐴)
 
11.2.5  The subalgebras of diagonal and scalar matrices

According to Wikipedia ("Diagonal Matrix", 8-Dec-2019, https://en.wikipedia.org/wiki/Diagonal_matrix): "In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices." The diagonal matrices are mentioned in [Lang] p. 576, but without giving them a dedicated definition. Furthermore, "A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple 𝜆 𝐼 of the identity matrix 𝐼. Its effect on a vector is scalar multiplication by 𝜆 [see scmatscm 20313!]". The scalar multiples of the identity matrix are mentioned in [Lang] p. 504, but without giving them a special name.

The main results of this subsection are the definitions of the sets of diagonal and scalar matrices (df-dmat 20290 and df-scmat 20291), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 20301), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 20320), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 20323) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 20321).

 
Syntaxcdmat 20288 Extend class notation for the algebra of diagonal matrices.
class DMat
 
Syntaxcscmat 20289 Extend class notation for the algebra of scalar matrices.
class ScMat
 
Definitiondf-dmat 20290* Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))})
 
Definitiondf-scmat 20291* Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))})
 
Theoremdmatval 20292* The set of 𝑁 x 𝑁 diagonal matrices over (a ring) 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
 
Theoremdmatel 20293* A 𝑁 x 𝑁 diagonal matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
 
Theoremdmatmat 20294 An 𝑁 x 𝑁 diagonal matrix over (the ring) 𝑅 is an 𝑁 x 𝑁 matrix over (the ring) 𝑅. (Contributed by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷𝑀𝐵))
 
Theoremdmatid 20295 The identity matrix is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) ∈ 𝐷)
 
Theoremdmatelnd 20296 An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )
 
Theoremdmatmul 20297* The product of two diagonal matrices. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )))
 
Theoremdmatsubcl 20298 The difference of two diagonal matrices is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(-g𝐴)𝑌) ∈ 𝐷)
 
Theoremdmatsgrp 20299 The set of diagonal matrices is a subgroup of the matrix group/algebra. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubGrp‘𝐴))
 
Theoremdmatmulcl 20300 The product of two diagonal matrices is a diagonal matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) ∈ 𝐷)
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
  Copyright terms: Public domain < Previous  Next >