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Type | Label | Description |
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Statement | ||
Definition | df-dmat 20701* | 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 20702* | 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 20703* | 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 20704* | 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 20705 | 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)) | ||
Theorem | dmatid 20706 | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝐷) | ||
Theorem | dmatelnd 20707 | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽)) → (𝐼𝑋𝐽) = 0 ) | ||
Theorem | dmatmul 20708* | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(.r‘𝐴)𝑌) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r‘𝑅)(𝑥𝑌𝑦)), 0 ))) | ||
Theorem | dmatsubcl 20709 | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(-g‘𝐴)𝑌) ∈ 𝐷) | ||
Theorem | dmatsgrp 20710 | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubGrp‘𝐴)) | ||
Theorem | dmatmulcl 20711 | 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 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷)) → (𝑋(.r‘𝐴)𝑌) ∈ 𝐷) | ||
Theorem | dmatsrng 20712 | The set of diagonal matrices is a subring of the matrix ring/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴)) | ||
Theorem | dmatcrng 20713 | The subring of diagonal matrices (over a commutative ring) is a commutative ring . (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝐷) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐶 ∈ CRing) | ||
Theorem | dmatscmcl 20714 | The multiplication of a diagonal matrix with a scalar is a diagonal matrix. (Contributed by AV, 19-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑀 ∈ 𝐷)) → (𝐶 ∗ 𝑀) ∈ 𝐷) | ||
Theorem | scmatval 20715* | The set of 𝑁 x 𝑁 scalar matrices over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )}) | ||
Theorem | scmatel 20716* | An 𝑁 x 𝑁 scalar matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))) | ||
Theorem | scmatscmid 20717* | A scalar matrix can be expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )) | ||
Theorem | scmatscmide 20718 | An entry of a scalar matrix expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐶 ∗ 1 )𝐽) = if(𝐼 = 𝐽, 𝐶, 0 )) | ||
Theorem | scmatscmiddistr 20719 | Distributive law for scalar and ring multiplication for scalar matrices expressed as multiplications of a scalar with the identity matrix. (Contributed by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆 ∈ 𝐵 ∧ 𝑇 ∈ 𝐵)) → ((𝑆 ∗ 1 ) × (𝑇 ∗ 1 )) = ((𝑆 · 𝑇) ∗ 1 )) | ||
Theorem | scmatmat 20720 | An 𝑁 x 𝑁 scalar 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) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐵)) | ||
Theorem | scmate 20721* | An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. (Contributed by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) | ||
Theorem | scmatmats 20722* | The set of an 𝑁 x 𝑁 scalar matrices over the ring 𝑅 expressed as a subset of 𝑁 x 𝑁 matrices over the ring 𝑅 with certain properties for their entries. (Contributed by AV, 31-Oct-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )}) | ||
Theorem | scmateALT 20723* | Alternate proof of scmate 20721: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 20722 but is longer and requires more distinct variables. (Contributed by AV, 19-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )) | ||
Theorem | scmatscm 20724* | The multiplication of a matrix with a scalar matrix corresponds to a scalar multiplication. (Contributed by AV, 28-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ × = (.r‘𝐴) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)) | ||
Theorem | scmatid 20725 | The identity matrix is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) ∈ 𝑆) | ||
Theorem | scmatdmat 20726 | A scalar matrix is a diagonal matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → 𝑀 ∈ 𝐷)) | ||
Theorem | scmataddcl 20727 | The sum of two scalar matrices is a scalar matrix. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐴)𝑌) ∈ 𝑆) | ||
Theorem | scmatsubcl 20728 | The difference of two scalar matrices is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(-g‘𝐴)𝑌) ∈ 𝑆) | ||
Theorem | scmatmulcl 20729 | The product of two scalar matrices is a scalar matrix. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(.r‘𝐴)𝑌) ∈ 𝑆) | ||
Theorem | scmatsgrp 20730 | The set of scalar matrices is a subgroup of the matrix group/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐴)) | ||
Theorem | scmatsrng 20731 | The set of scalar matrices is a subring of the matrix ring/algebra. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐴)) | ||
Theorem | scmatcrng 20732 | The subring of scalar matrices (over a commutative ring) is a commutative ring. (Contributed by AV, 21-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CRing) | ||
Theorem | scmatsgrp1 20733 | The set of scalar matrices is a subgroup of the group/ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝐷) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) | ||
Theorem | scmatsrng1 20734 | The set of scalar matrices is a subring of the ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐸 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ 𝐷 = (𝑁 DMat 𝑅) & ⊢ 𝐶 = (𝐴 ↾s 𝐷) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) | ||
Theorem | smatvscl 20735 | Closure of the scalar multiplication in the ring of scalar matrices. (matvscl 20641 analog.) (Contributed by AV, 24-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) & ⊢ ∗ = ( ·𝑠 ‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑆)) → (𝐶 ∗ 𝑋) ∈ 𝑆) | ||
Theorem | scmatlss 20736 | The set of scalar matrices is a linear subspace of the matrix algebra. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (LSubSp‘𝐴)) | ||
Theorem | scmatstrbas 20737 | The set of scalar matrices is the base set of the ring of corresponding scalar matrices. (Contributed by AV, 26-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑆) = 𝐶) | ||
Theorem | scmatrhmval 20738* | The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝑋 ∗ 1 )) | ||
Theorem | scmatrhmcl 20739* | The value of the ring homomorphism 𝐹 is a scalar matrix. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) ∈ 𝐶) | ||
Theorem | scmatf 20740* | There is a function from a ring to any ring of scalar matrices over this ring. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾⟶𝐶) | ||
Theorem | scmatfo 20741* | There is a function from a ring onto any ring of scalar matrices over this ring. (Contributed by AV, 26-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–onto→𝐶) | ||
Theorem | scmatf1 20742* | There is a 1-1 function from a ring to any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 25-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1→𝐶) | ||
Theorem | scmatf1o 20743* | There is a bijection between a ring and any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 26-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾–1-1-onto→𝐶) | ||
Theorem | scmatghm 20744* | There is a group homomorphism from the additive group of a ring to the additive group of the ring of scalar matrices over this ring. (Contributed by AV, 22-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | ||
Theorem | scmatmhm 20745* | There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑇 = (mulGrp‘𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑀 MndHom 𝑇)) | ||
Theorem | scmatrhm 20746* | There is a ring homomorphism from a ring to the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | scmatrngiso 20747* | There is a ring isomorphism from a ring to the ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ∗ 1 )) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingIso 𝑆)) | ||
Theorem | scmatric 20748 | A ring is isomorphic to every ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐶 = (𝑁 ScMat 𝑅) & ⊢ 𝑆 = (𝐴 ↾s 𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝑅 ≃𝑟 𝑆) | ||
Theorem | mat0scmat 20749 | The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 20726, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.) |
⊢ (𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅)) | ||
Theorem | mat1scmat 20750 | A 1-dimensional matrix over a ring is always a scalar matrix (and therefore, by scmatdmat 20726, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ 𝑉 ∧ (♯‘𝑁) = 1 ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝐵 → 𝑀 ∈ (𝑁 ScMat 𝑅))) | ||
The module of 𝑛-dimensional "column vectors" over a ring 𝑟 is the 𝑛-dimensional free module over a ring 𝑟, which is the product of 𝑛 -many copies of the ring with componentwise addition and multiplication. Although a "column vector" could also be defined as n x 1 -matrix (according to Wikipedia "Row and column vectors", 22-Feb-2019, https://en.wikipedia.org/wiki/Row_and_column_vectors: "In linear algebra, a column vector [... ] is an m x 1 matrix, that is, a matrix consisting of a single column of m elements"), which would allow for using the matrix multiplication df-mamu 20594 for multiplying a matrix with a column vector, it seems more natural to use the definition of a free (left) module, avoiding to provide a singleton as 1-dimensional index set for the column, and to introduce a new operator df-mvmul 20752 for the multiplication of a matrix with a column vector. In most cases, it is sufficient to regard members of ((Base‘𝑅) ↑𝑚 𝑁) as "column vectors", because ((Base‘𝑅) ↑𝑚 𝑁) is the base set of (𝑅 freeLMod 𝑁), see frlmbasmap 20502. See also the statements in [Lang] p. 508. | ||
Syntax | cmvmul 20751 | Syntax for the operator for the multiplication of a vector with a matrix. |
class maVecMul | ||
Definition | df-mvmul 20752* | The operator which multiplies an M x N -matrix with an N-dimensional vector. (Contributed by AV, 23-Feb-2019.) |
⊢ maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd ‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))))) | ||
Theorem | mvmulfval 20753* | Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.) |
⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) ⇒ ⊢ (𝜑 → × = (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦‘𝑗))))))) | ||
Theorem | mvmulval 20754* | Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.) |
⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑀 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) | ||
Theorem | mvmulfv 20755* | A cell/element in the vector resulting from a multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.) |
⊢ × = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝐼 ∈ 𝑀) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) | ||
Theorem | mavmulval 20756* | Multiplication of a vector with a square matrix. (Contributed by AV, 23-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))) | ||
Theorem | mavmulfv 20757* | A cell/element in the vector resulting from a multiplication of a vector with a square matrix. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 18-Feb-2019.) (Revised by AV, 23-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌‘𝑗))))) | ||
Theorem | mavmulcl 20758 | Multiplication of an NxN matrix with an N-dimensional vector results in an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑𝑚 𝑁)) | ||
Theorem | 1mavmul 20759 | Multiplication of the identity NxN matrix with an N-dimensional vector results in the vector itself. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 23-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) ⇒ ⊢ (𝜑 → ((1r‘𝐴) · 𝑌) = 𝑌) | ||
Theorem | mavmulass 20760 | Associativity of the multiplication of two NxN matrices with an N-dimensional vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 25-Feb-2019.) (Proof shortened by AV, 22-Jul-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ × = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴)) ⇒ ⊢ (𝜑 → ((𝑋 × 𝑍) · 𝑌) = (𝑋 · (𝑍 · 𝑌))) | ||
Theorem | mavmuldm 20761 | The domain of the matrix vector multiplication function. (Contributed by AV, 27-Feb-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (𝐵 ↑𝑚 (𝑀 × 𝑁)) & ⊢ 𝐷 = (𝐵 ↑𝑚 𝑁) & ⊢ · = (𝑅 maVecMul 〈𝑀, 𝑁〉) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → dom · = (𝐶 × 𝐷)) | ||
Theorem | mavmulsolcl 20762 | Every solution of the equation 𝐴∗𝑋 = 𝑌 for a matrix 𝐴 and a vector 𝐵 is a vector. (Contributed by AV, 27-Feb-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (𝐵 ↑𝑚 (𝑀 × 𝑁)) & ⊢ 𝐷 = (𝐵 ↑𝑚 𝑁) & ⊢ · = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐸 = (𝐵 ↑𝑚 𝑀) ⇒ ⊢ (((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑀 ≠ ∅) ∧ (𝑅 ∈ 𝑉 ∧ 𝑌 ∈ 𝐸)) → ((𝐴 · 𝑋) = 𝑌 → 𝑋 ∈ 𝐷)) | ||
Theorem | mavmul0 20763 | Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.) |
⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) ⇒ ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) | ||
Theorem | mavmul0g 20764 | The result of the 0-dimensional multiplication of a matrix with a vector is always the empty set. (Contributed by AV, 1-Mar-2019.) |
⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) ⇒ ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑋 · 𝑌) = ∅) | ||
Theorem | mvmumamul1 20765* | The multiplication of an MxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an MxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.) |
⊢ × = (𝑅 maMul 〈𝑀, 𝑁, {∅}〉) & ⊢ · = (𝑅 maVecMul 〈𝑀, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ Fin) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁))) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑𝑚 (𝑁 × {∅}))) ⇒ ⊢ (𝜑 → (∀𝑗 ∈ 𝑁 (𝑌‘𝑗) = (𝑗𝑍∅) → ∀𝑖 ∈ 𝑀 ((𝐴 · 𝑌)‘𝑖) = (𝑖(𝐴 × 𝑍)∅))) | ||
Theorem | mavmumamul1 20766* | The multiplication of an NxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an NxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ × = (𝑅 maMul 〈𝑁, 𝑁, {∅}〉) & ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 𝑁)) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ↑𝑚 (𝑁 × {∅}))) ⇒ ⊢ (𝜑 → (∀𝑗 ∈ 𝑁 (𝑌‘𝑗) = (𝑗𝑍∅) → ∀𝑖 ∈ 𝑁 ((𝑋 · 𝑌)‘𝑖) = (𝑖(𝑋 × 𝑍)∅))) | ||
Syntax | cmarrep 20767 | Syntax for the row replacing function for a square matrix. |
class matRRep | ||
Syntax | cmatrepV 20768 | Syntax for the function replacing a column of a matrix by a vector. |
class matRepV | ||
Definition | df-marrep 20769* | Define the matrices whose k-th row is replaced by 0's and an arbitrary element of the underlying ring at the l-th column. (Contributed by AV, 12-Feb-2019.) |
⊢ matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗)))))) | ||
Definition | df-marepv 20770* | Function replacing a column of a matrix by a vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) | ||
Theorem | marrepfval 20771* | First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))))) | ||
Theorem | marrepval0 20772* | Second substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))) | ||
Theorem | marrepval 20773* | Third substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗)))) | ||
Theorem | marrepeval 20774 | An entry of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRRep 𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽))) | ||
Theorem | marrepcl 20775 | Closure of the row replacement function for square matrices. (Contributed by AV, 13-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) ∈ 𝐵) | ||
Theorem | marepvfval 20776* | First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗))))) | ||
Theorem | marepvval0 20777* | Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗))))) | ||
Theorem | marepvval 20778* | Third substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)))) | ||
Theorem | marepveval 20779 | An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑄 = (𝑁 matRepV 𝑅) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼𝑀𝐽))) | ||
Theorem | marepvcl 20780 | Closure of the column replacement function for square matrices. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) | ||
Theorem | ma1repvcl 20781 | Closure of the column replacement function for identity matrices. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) | ||
Theorem | ma1repveval 20782 | An entry of an identity matrix with a replaced column. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), if(𝐽 = 𝐼, (1r‘𝑅), 0 ))) | ||
Theorem | mulmarep1el 20783 | Element by element multiplication of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → ((𝐼𝑋𝐿)(.r‘𝑅)(𝐿𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝐿)(.r‘𝑅)(𝐶‘𝐿)), if(𝐽 = 𝐿, (𝐼𝑋𝐿), 0 ))) | ||
Theorem | mulmarep1gsum1 20784* | The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 ≠ 𝐾)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽)) | ||
Theorem | mulmarep1gsum2 20785* | The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 18-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘𝐴) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) & ⊢ × = (𝑅 maVecMul 〈𝑁, 𝑁〉) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑅 Σg (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.r‘𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍‘𝐼), (𝐼𝑋𝐽))) | ||
Theorem | 1marepvmarrepid 20786 | Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.) |
⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) & ⊢ 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍‘𝐼))𝐼) = 𝑋) | ||
Syntax | csubma 20787 | Syntax for submatrices of a square matrix. |
class subMat | ||
Definition | df-subma 20788* | Define the submatrices of a square matrix. A submatrix is obtained by deleting a row and a column of the original matrix. Since the indices of a matrix need not to be sequential integers, it does not matter that there may be gaps in the numbering of the indices for the submatrix. The determinants of such submatrices are called the "minors" of the original matrix. (Contributed by AV, 27-Dec-2018.) |
⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) | ||
Theorem | submabas 20789* | Any subset of the index set of a square matrix defines a submatrix of the matrix. (Contributed by AV, 1-Jan-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐷 ⊆ 𝑁) → (𝑖 ∈ 𝐷, 𝑗 ∈ 𝐷 ↦ (𝑖𝑀𝑗)) ∈ (Base‘(𝐷 Mat 𝑅))) | ||
Theorem | submafval 20790* | First substitution for a submatrix. (Contributed by AV, 28-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (𝑁 subMat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ 𝑄 = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))) | ||
Theorem | submaval0 20791* | Second substitution for a submatrix. (Contributed by AV, 28-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (𝑁 subMat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑀 ∈ 𝐵 → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗)))) | ||
Theorem | submaval 20792* | Third substitution for a submatrix. (Contributed by AV, 28-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (𝑁 subMat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗))) | ||
Theorem | submaeval 20793 | An entry of a submatrix of a square matrix. (Contributed by AV, 28-Dec-2018.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (𝑁 subMat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ (𝑁 ∖ {𝐾}) ∧ 𝐽 ∈ (𝑁 ∖ {𝐿}))) → (𝐼(𝐾(𝑄‘𝑀)𝐿)𝐽) = (𝐼𝑀𝐽)) | ||
Theorem | 1marepvsma1 20794 | The submatrix of the identity matrix with the ith column replaced by the vector obtained by removing the ith row and the ith column is an identity matrix. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.) |
⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) & ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) & ⊢ 𝑋 = (( 1 (𝑁 matRepV 𝑅)𝑍)‘𝐼) ⇒ ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) → (𝐼((𝑁 subMat 𝑅)‘𝑋)𝐼) = (1r‘((𝑁 ∖ {𝐼}) Mat 𝑅))) | ||
Syntax | cmdat 20795 | Syntax for the matrix determinant function. |
class maDet | ||
Definition | df-mdet 20796* | Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 20798). The properties of the axiomatic definition of a determinant according to [Weierstrass] p. 272 are derived from this definition as theorems: "The determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring". The functionality is shown by mdetf 20806. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 20813, the homogeneity by mdetrsca 20814. Furthermore, it is shown that the determinant function is alternating (see mdetralt 20819) and normalized (see mdet1 20812). Finally, the uniqueness is shown by mdetuni 20833. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 20798. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 10-Jul-2018.) |
⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))) | ||
Theorem | mdetfval 20797* | First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.) |
⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (mulGrp‘𝑅) ⇒ ⊢ 𝐷 = (𝑚 ∈ 𝐵 ↦ (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) | ||
Theorem | mdetleib 20798* | Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by SO, 9-Jul-2018.) |
⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (mulGrp‘𝑅) ⇒ ⊢ (𝑀 ∈ 𝐵 → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑀𝑥))))))) | ||
Theorem | mdetleib2 20799* | Leibniz' formula can also be expanded by rows. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.) |
⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑌 = (ℤRHom‘𝑅) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (mulGrp‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (𝑅 Σg (𝑝 ∈ 𝑃 ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝑈 Σg (𝑥 ∈ 𝑁 ↦ (𝑥𝑀(𝑝‘𝑥)))))))) | ||
Theorem | nfimdetndef 20800 | The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.) |
⊢ 𝐷 = (𝑁 maDet 𝑅) ⇒ ⊢ (𝑁 ∉ Fin → 𝐷 = ∅) |
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