| Metamath
Proof Explorer Theorem List (p. 225 of 494) | < 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: | (1-30937) |
(30938-32460) |
(32461-49324) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mamures 22401 | Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.) |
| ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ 𝐺 = (𝑅 maMul 〈𝐼, 𝑁, 𝑃〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝐼 ⊆ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌)) | ||
| Theorem | grpvlinv 22402 | Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → ((𝑁 ∘ 𝑋) ∘f + 𝑋) = (𝐼 × { 0 })) | ||
| Theorem | grpvrinv 22403 | Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f + (𝑁 ∘ 𝑋)) = (𝐼 × { 0 })) | ||
| Theorem | ringvcl 22404 | Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ↑m 𝐼) ∧ 𝑌 ∈ (𝐵 ↑m 𝐼)) → (𝑋 ∘f · 𝑌) ∈ (𝐵 ↑m 𝐼)) | ||
| Theorem | mamucl 22405 | Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑃〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵 ↑m (𝑀 × 𝑃))) | ||
| Theorem | mamuass 22406 | Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (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))) | ||
| Theorem | mamudi 22407 | 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) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ + = (+g‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → ((𝑋 ∘f + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘f + (𝑌𝐹𝑍))) | ||
| Theorem | mamudir 22408 | 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) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ + = (+g‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑂))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → (𝑋𝐹(𝑌 ∘f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍))) | ||
| Theorem | mamuvs1 22409 | Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍))) | ||
| Theorem | mamuvs2 22410 | Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.) |
| ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑁, 𝑂〉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑m (𝑁 × 𝑂))) ⇒ ⊢ (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘f · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘f · (𝑋𝐹𝑍))) | ||
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 22435. 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. | ||
| Syntax | cmat 22411 | Syntax for the square matrix algebra. |
| class Mat | ||
| Definition | df-mat 22412* | 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〉)〉)) | ||
| Theorem | matbas0pc 22413 | 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)) = ∅) | ||
| Theorem | matbas0 22414 | 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)) = ∅) | ||
| Theorem | matval 22415 | 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 ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet 〈(.r‘ndx), · 〉)) | ||
| Theorem | matrcl 22416 | Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) | ||
| Theorem | matbas 22417 | 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 ∧ 𝑅 ∈ 𝑉) → (Base‘𝐺) = (Base‘𝐴)) | ||
| Theorem | matplusg 22418 | 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 ∧ 𝑅 ∈ 𝑉) → (+g‘𝐺) = (+g‘𝐴)) | ||
| Theorem | matsca 22419 | The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴)) | ||
| Theorem | matscaOLD 22420 | Obsolete version of matsca 22419 as of 12-Nov-2024. The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴)) | ||
| Theorem | matvsca 22421 | The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴)) | ||
| Theorem | matvscaOLD 22422 | Obsolete version of matvsca 22421 as of 12-Nov-2024. The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴)) | ||
| Theorem | mat0 22423 | 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 ∧ 𝑅 ∈ 𝑉) → (0g‘𝐺) = (0g‘𝐴)) | ||
| Theorem | matinvg 22424 | 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 ∧ 𝑅 ∈ 𝑉) → (invg‘𝐺) = (invg‘𝐴)) | ||
| Theorem | mat0op 22425* | Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 )) | ||
| Theorem | matsca2 22426 | The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 = (Scalar‘𝐴)) | ||
| Theorem | matbas2 22427 | 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 ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) | ||
| Theorem | matbas2i 22428 | A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (𝐾 ↑m (𝑁 × 𝑁))) | ||
| Theorem | matbas2d 22429* | 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) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵) | ||
| Theorem | eqmat 22430* | 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) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗))) | ||
| Theorem | matecl 22431 | 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) ⇒ ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾) | ||
| Theorem | matecld 22432 | 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) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) & ⊢ (𝜑 → 𝐽 ∈ 𝑁) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾) | ||
| Theorem | matplusg2 22433 | Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ✚ = (+g‘𝐴) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ✚ 𝑌) = (𝑋 ∘f + 𝑌)) | ||
| Theorem | matvsca2 22434 | 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) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ × = (.r‘𝑅) & ⊢ 𝐶 = (𝑁 × 𝑁) ⇒ ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘f × 𝑌)) | ||
| Theorem | matlmod 22435 | 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) | ||
| Theorem | matgrp 22436 | The matrix ring is a group. (Contributed by AV, 21-Dec-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp) | ||
| Theorem | matvscl 22437 | Closure of the scalar multiplication in the matrix ring. (lmodvscl 20876 analog.) (Contributed by AV, 27-Nov-2019.) |
| ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵)) → (𝐶 · 𝑋) ∈ 𝐵) | ||
| Theorem | matsubg 22438 | 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 ∧ 𝑅 ∈ 𝑉) → (-g‘𝐺) = (-g‘𝐴)) | ||
| Theorem | matplusgcell 22439 | Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ✚ = (+g‘𝐴) & ⊢ + = (+g‘𝑅) ⇒ ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 ✚ 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽))) | ||
| Theorem | matsubgcell 22440 | Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (-g‘𝐴) & ⊢ − = (-g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) − (𝐼𝑌𝐽))) | ||
| Theorem | matinvgcell 22441 | Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = (invg‘𝑅) & ⊢ 𝑊 = (invg‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑊‘𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽))) | ||
| Theorem | matvscacell 22442 | 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) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ × = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽))) | ||
| Theorem | matgsum 22443* | Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ 𝐵) & ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) | ||
The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 22449) and an associative algebra (see matassa 22450). Additionally, theorems for the identity matrix and transposed matrices are provided. | ||
| Theorem | matmulr 22444 | Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → · = (.r‘𝐴)) | ||
| Theorem | mamumat1cl 22445* | The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) ⇒ ⊢ (𝜑 → 𝐼 ∈ (𝐵 ↑m (𝑀 × 𝑀))) | ||
| Theorem | mat1comp 22446* | The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) ⇒ ⊢ ((𝐴 ∈ 𝑀 ∧ 𝐽 ∈ 𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 )) | ||
| Theorem | mamulid 22447* | 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) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ 𝐹 = (𝑅 maMul 〈𝑀, 𝑀, 𝑁〉) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) ⇒ ⊢ (𝜑 → (𝐼𝐹𝑋) = 𝑋) | ||
| Theorem | mamurid 22448* | 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) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (𝑖 ∈ 𝑀, 𝑗 ∈ 𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 )) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ 𝐹 = (𝑅 maMul 〈𝑁, 𝑀, 𝑀〉) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑁 × 𝑀))) ⇒ ⊢ (𝜑 → (𝑋𝐹𝐼) = 𝑋) | ||
| Theorem | matring 22449 | 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) | ||
| Theorem | matassa 22450 | 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) | ||
| Theorem | matmulcell 22451* | Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) (Revised by AV, 3-Jul-2022.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ × = (.r‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(.r‘𝑅)(𝑗𝑌𝐽))))) | ||
| Theorem | mpomatmul 22452* | 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) & ⊢ × = (.r‘𝐴) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ 𝑋 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐶) & ⊢ 𝑌 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝐸) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐸 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑘 = 𝑖 ∧ 𝑚 = 𝑗)) → 𝐷 = 𝐶) & ⊢ ((𝜑 ∧ (𝑚 = 𝑖 ∧ 𝑙 = 𝑗)) → 𝐹 = 𝐸) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁 ∧ 𝑚 ∈ 𝑁) → 𝐷 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 𝐹 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑅 Σg (𝑚 ∈ 𝑁 ↦ (𝐷 · 𝐹))))) | ||
| Theorem | mat1 22453* | 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 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) | ||
| Theorem | mat1ov 22454 | Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) & ⊢ (𝜑 → 𝐽 ∈ 𝑁) & ⊢ 𝑈 = (1r‘𝐴) ⇒ ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 )) | ||
| Theorem | mat1bas 22455 | 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) → 1 ∈ 𝐵) | ||
| Theorem | matsc 22456* | The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐾) → (𝐿 · (1r‘𝐴)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 ))) | ||
| Theorem | ofco2 22457 | 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 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻))) | ||
| Theorem | oftpos 22458 | 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.) |
| ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → tpos (𝐹 ∘f 𝑅𝐺) = (tpos 𝐹 ∘f 𝑅tpos 𝐺)) | ||
| Theorem | mattposcl 22459 | 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) ⇒ ⊢ (𝑀 ∈ 𝐵 → tpos 𝑀 ∈ 𝐵) | ||
| Theorem | mattpostpos 22460 | 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) ⇒ ⊢ (𝑀 ∈ 𝐵 → tpos tpos 𝑀 = 𝑀) | ||
| Theorem | mattposvs 22461 | 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) & ⊢ · = ( ·𝑠 ‘𝐴) ⇒ ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌)) | ||
| Theorem | mattpos1 22462 | The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 ) | ||
| Theorem | tposmap 22463 | 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 ∈ (𝐵 ↑m (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵 ↑m (𝐽 × 𝐼))) | ||
| Theorem | mamutpos 22464 | 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) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑m (𝑁 × 𝑃))) ⇒ ⊢ (𝜑 → tpos (𝑋𝐹𝑌) = (tpos 𝑌𝐺tpos 𝑋)) | ||
| Theorem | mattposm 22465 | 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) & ⊢ · = (.r‘𝐴) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋)) | ||
| Theorem | matgsumcl 22466* | 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 ∧ 𝑀 ∈ 𝐵) → (𝑈 Σg (𝑟 ∈ 𝑁 ↦ (𝑟𝑀𝑟))) ∈ (Base‘𝑅)) | ||
| Theorem | madetsumid 22467* | 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) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑃 = ( I ↾ 𝑁)) → (((𝑌 ∘ 𝑆)‘𝑃) · (𝑈 Σg (𝑟 ∈ 𝑁 ↦ ((𝑃‘𝑟)𝑀𝑟)))) = (𝑈 Σg (𝑟 ∈ 𝑁 ↦ (𝑟𝑀𝑟)))) | ||
| Theorem | matepmcl 22468* | 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 ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 ((𝑄‘𝑛)𝑀𝑛) ∈ (Base‘𝑅)) | ||
| Theorem | matepm2cl 22469* | 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 ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → ∀𝑛 ∈ 𝑁 (𝑛𝑀(𝑄‘𝑛)) ∈ (Base‘𝑅)) | ||
| Theorem | madetsmelbas 22470* | 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 ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛)))) ∈ (Base‘𝑅)) | ||
| Theorem | madetsmelbas2 22471* | 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 ∧ 𝑀 ∈ 𝐵 ∧ 𝑄 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛𝑀(𝑄‘𝑛))))) ∈ (Base‘𝑅)) | ||
As already mentioned before, and shown in mat0dimbas0 22472, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). 22472 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 22476. 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 22493. | ||
| Theorem | mat0dimbas0 22472 | The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.) |
| ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) | ||
| Theorem | mat0dim0 22473 | The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
| ⊢ 𝐴 = (∅ Mat 𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (0g‘𝐴) = ∅) | ||
| Theorem | mat0dimid 22474 | The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.) |
| ⊢ 𝐴 = (∅ Mat 𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (1r‘𝐴) = ∅) | ||
| Theorem | mat0dimscm 22475 | 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( ·𝑠 ‘𝐴)∅) = ∅) | ||
| Theorem | mat0dimcrng 22476 | 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) | ||
| Theorem | mat1dimelbas 22477* | 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 ∧ 𝐸 ∈ 𝑉) → (𝑀 ∈ (Base‘𝐴) ↔ ∃𝑟 ∈ 𝐵 𝑀 = {〈𝑂, 𝑟〉})) | ||
| Theorem | mat1dimbas 22478 | 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 ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → {〈𝑂, 𝑋〉} ∈ (Base‘𝐴)) | ||
| Theorem | mat1dim0 22479 | 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 ∧ 𝐸 ∈ 𝑉) → (0g‘𝐴) = {〈𝑂, (0g‘𝑅)〉}) | ||
| Theorem | mat1dimid 22480 | 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 ∧ 𝐸 ∈ 𝑉) → (1r‘𝐴) = {〈𝑂, (1r‘𝑅)〉}) | ||
| Theorem | mat1dimscm 22481 | 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 ∧ 𝐸 ∈ 𝑉) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝐴){〈𝑂, 𝑌〉}) = {〈𝑂, (𝑋(.r‘𝑅)𝑌)〉}) | ||
| Theorem | mat1dimmul 22482 | 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 ∧ 𝐸 ∈ 𝑉) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ({〈𝑂, 𝑋〉} (.r‘𝐴){〈𝑂, 𝑌〉}) = {〈𝑂, (𝑋(.r‘𝑅)𝑌)〉}) | ||
| Theorem | mat1dimcrng 22483 | 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 ∧ 𝐸 ∈ 𝑉) → 𝐴 ∈ CRing) | ||
| Theorem | mat1f1o 22484* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾–1-1-onto→𝐵) | ||
| Theorem | mat1rhmval 22485* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = {〈𝑂, 𝑋〉}) | ||
| Theorem | mat1rhmelval 22486* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐸(𝐹‘𝑋)𝐸) = 𝑋) | ||
| Theorem | mat1rhmcl 22487* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) ∈ 𝐵) | ||
| Theorem | mat1f 22488* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹:𝐾⟶𝐵) | ||
| Theorem | mat1ghm 22489* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 GrpHom 𝐴)) | ||
| Theorem | mat1mhm 22490* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑀 MndHom 𝑁)) | ||
| Theorem | mat1rhm 22491* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 RingHom 𝐴)) | ||
| Theorem | mat1rngiso 22492* | 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 = (𝑥 ∈ 𝐾 ↦ {〈𝑂, 𝑥〉}) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐸 ∈ 𝑉) → 𝐹 ∈ (𝑅 RingIso 𝐴)) | ||
| Theorem | mat1ric 22493 | 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 ∧ 𝐸 ∈ 𝑉) → 𝑅 ≃𝑟 𝐴) | ||
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 22519!]". 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 22496 and df-scmat 22497), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices form a subring of the ring of square matrices (dmatsrng 22507), that the scalar matrices form a subring of the ring of square matrices (scmatsrng 22526), that the scalar matrices form a subring of the ring of diagonal matrices (scmatsrng1 22529) and that the ring of scalar matrices over a commutative ring is a commutative ring (scmatcrng 22527). | ||
| Syntax | cdmat 22494 | Extend class notation for the algebra of diagonal matrices. |
| class DMat | ||
| Syntax | cscmat 22495 | Extend class notation for the algebra of scalar matrices. |
| class ScMat | ||
| Definition | df-dmat 22496* | 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)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) | ||
| Definition | df-scmat 22497* | 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( ·𝑠 ‘𝑎)(1r‘𝑎))}) | ||
| Theorem | dmatval 22498* | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) | ||
| Theorem | dmatel 22499* | A 𝑁 x 𝑁 diagonal matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 ↔ (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )))) | ||
| Theorem | dmatmat 22500 | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝐷 → 𝑀 ∈ 𝐵)) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |