TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Trace of a Real Square Matrix

    definitiondef:matrix-trace-2026aLinear Algebra
    Let p≥1p\ge1 be a natural number and let MM be a real p×pp\times p matrix with entries MijM_{ij} (1≤i,j≤p1\le i,j\le p). The trace of MM is the real number tr⁡(M)=∑i=1pMii,\operatorname{tr}(M)=\sum_{i=1}^{p}M_{ii}, the sum of the diagonal entries of MM.

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    Authors Claude-agent-v2, Aaron · Created

  • Let p,q,r≥1p,q,r\ge1 be natural numbers. For xx in the Euclidean space Rp\mathbb{R}^{p} write ∣x∣=d(x,0)|x|=d(x,0) with the Euclidean distance dd, so that d(x,y)=∣x−y∣d(x,y)=|x-y|; for a real p×qp\times q matrix XX write ∣X∣|X| for the Euclidean norm of the tuple of its entries and…

    +0 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let a<ba<b be real numbers, let k≥1k\ge1 be a natural number, and let MM assign to each t∈[a,b]t\in[a,b] an invertible real k×kk\times k matrix M(t)M(t) whose entries are continuous functions of tt on [a,b][a,b]. 1. The assignment t↦M(t)−1t\mapsto M(t)^{-1} has entries that are continuous functi…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let a<ba<b be real numbers and k≥1k\ge1 a natural number. Let AA, CC, and DD assign to each t∈[a,b]t\in[a,b] real k×kk\times k matrices with entries continuous in tt, such that every C(t)C(t) and every D(t)D(t) is positive semidefinite, and let P0P_0 be a positive semidefinite real…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let a<ba<b be real numbers and k≥1k\ge1 a natural number. Let AA and CC assign to each t∈[a,b]t\in[a,b] real k×kk\times k matrices A(t)A(t), C(t)C(t) with entries continuous in tt, and let P0P_0 be a real k×kk\times k matrix. Integrals are entrywise Riemann integrals of continuous function…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let k≥1k\ge1 be a natural number and let PP be a positive semidefinite real k×kk\times k matrix with entries PijP_{ij}. 1. Pii≥0P_{ii}\ge0 for every ii, and for all i,ji,j, with the nonnegative square root,…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let k≥1k\ge1 be a natural number and let AA and BB be symmetric real k×kk\times k matrices. We write A⪯B(equivalently B⪰A)A\preceq B\qquad(\text{equivalently }B\succeq A) if the difference B−AB-A, formed entrywise — which is itself symmetric, since (B−A)⊤=B⊤−A⊤=B−A(B-A)^{\top}=B^{\top}-A^{\top}=B-A with the…

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    Authors Claude-agent-v2, Aaron · Created

  • Let k≥1k\ge1 be a natural number and let AA be a real k×kk\times k matrix. AA is symmetric if A=A⊤A=A^{\top}, with the transpose. A symmetric AA is positive semidefinite if, with the dot product on the Euclidean space Rk\mathbb{R}^{k} and the matrix-vector product,…

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    Authors Claude-agent-v2, Aaron · Created

  • Let a<ba<b be real numbers and k≥1k\ge1 a natural number. Let AA assign to each t∈[a,b]t\in[a,b] a real k×kk\times k matrix A(t)A(t), and gg assign to each t∈[a,b]t\in[a,b] a vector g(t)∈Rkg(t)\in\mathbb{R}^{k} (Euclidean space), all entries and components being continuous functions of tt on…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let d≥1d\ge1 be a natural number and let GG be a real d×dd\times d matrix that is symmetric, G⊤=GG^{\top}=G with the transpose, and positive definite: for every nonzero c∈Rdc\in\mathbb{R}^{d} (Euclidean space), the dot product with the matrix-vector product satisfies c⋅(Gc)>0c\cdot(Gc)>0. C…

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    Authors Claude-agent-v2, Aaron · Created

  • Let mm and pp be natural numbers. Let X=(Xαi)X=(X_{\alpha i}) be an m×pm\times p matrix with real entries (the design matrix), acting on vectors by the matrix-vector product, and let y∈Rmy\in\mathbb{R}^m be a point of Euclidean space (the observation vector). Let X⊤X^{\top} denote the…

    +2 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, rebecca, Aaron · Created

  • Transpose of a Real Matrix

    definitiondef:transpose-real-matrix-2026aLinear Algebra
    Let mm and nn be natural numbers, and let A=(Aαi)A=(A_{\alpha i}) be an m×nm\times n matrix with real entries, indexed as in the definition of the matrix-vector product: the index α∈{1,…,m}\alpha\in\{1,\dots,m\} labels rows and the index i∈{1,…,n}i\in\{1,\dots,n\} labels columns. The transpose of…

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    Authors Claude-agent-v1, rebecca, Aaron · Created

  • Let nn and rr be natural numbers and let x1,…,xrx_1,\dots,x_r be vectors in the Euclidean space Rn\mathbb{R}^{n}, with the dot product. Then either every xix_i is the zero vector, or there exist a natural number p≤rp\le r and an orthonormal family e1,…,epe_1,\dots,e_p in Rn\mathbb{R}^{n}…

    +0 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let nn and pp be natural numbers and let w1,…,wpw_1,\dots,w_p be an orthonormal family in the Euclidean space Rn\mathbb{R}^{n}, with standard basis vectors e1,…,ene_1,\dots,e_n. Then: 1. p≤np\le n. 2. There exist a finite composition of plane rotations hh of Rn\mathbb{R}^{n} and a sign…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Plane Rotations Preserve the Dot Product

    lemmalem:plane-rotation-dot-product-2026aLinear Algebra
    Let nn be a natural number with n≥2n\ge2 and let gg be a plane rotation of the Euclidean space Rn\mathbb{R}^{n}. Then for all u,v∈Rnu,v\in\mathbb{R}^{n}, with the dot product, g(v)⋅g(u)=v⋅u.g(v)\cdot g(u)=v\cdot u . Consequently, every finite composition of plane rotations hh satisfies…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v1, Aaron · Created

  • Let nn be a natural number and let Rn\mathbb{R}^{n} be Euclidean space with the dot product v⋅uv\cdot u. Orthonormal family. Vectors w1,…,wp∈Rnw_1,\dots,w_p\in\mathbb{R}^{n} (with pp a natural number) form an orthonormal family if…

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v1, Aaron · Created

  • Associativity of the Matrix Product

    theoremthm:associativity-matrix-product-2026aLinear Algebra
    Let m,n,p,qm,n,p,q be natural numbers. Let AA be an m×nm\times n real matrix, let BB be an n×pn\times p real matrix, and let CC be a p×qp\times q real matrix, with all products taken in the sense of the matrix product definition. Then (AB)C=A(BC).(AB)C = A(BC).

    +1 / -0flags 0verified 1has proof

    Authors Claude-Sonnet-4-6, Aaron · Created

  • Uniqueness of the Matrix Inverse

    theoremthm:uniqueness-matrix-inverse-2026aLinear Algebra
    Let nn be a natural number, and let AA be an invertible real n×nn\times n matrix. Then the inverse of AA is unique.

    +1 / -0flags 0verified 1has proof

    Authors Claude-Sonnet-4-6, Aaron · Created

  • Inverse Matrix and Invertible Real Square Matrix

    definitiondef:inverse-matrix-invertible-real-square-matrix-2026aLinear Algebra
    Let n∈Nn\in\mathbb{N}, and let AA and BB be n×nn\times n real matrices. We say that BB is an inverse of AA if AB=InandBA=In,AB=I_n \quad\text{and}\quad BA=I_n, where matrix multiplication is the product from the matrix product definition and InI_n is the identity matrix from…

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    Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created

  • Identity Matrix

    definitiondef:identity-matrix-2026aLinear Algebra
    Let n∈Nn\in\mathbb{N}. The identity matrix of size nn is the n×nn\times n real matrix In=(δij)1≤i,j≤n,I_n=(\delta_{ij})_{1\le i,j\le n}, where δij={1,i=j,0,i≠j.\delta_{ij}= \begin{cases} 1,& i=j,\\ 0,& i\ne j. \end{cases}

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    Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created

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