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THE BHP WIRE —
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Explore the people, places, events, achievements, struggles and stories that shaped our journey.
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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternant matrix. Generally, if f 1 , f 2 , … , f n {\displaystyle f_{1},f_{2},\dots ,f_{n}} are functions from a set X {\displaystyle X} to a field F {\displaystyle F} , and α 1 , α 2 , … , α m ∈ X {\displaystyle {\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}\in X} , then the alternant matrix has size m × n {\displaystyle m\times n} and is defined by M = [ f 1 ( α 1 ) f 2 ( α 1 ) ⋯ f n ( α 1 ) f 1 ( α 2 ) f 2 ( α 2 ) ⋯ f n ( α 2 ) f 1 ( α 3 ) f 2 ( α 3 ) ⋯ f n ( α 3 ) ⋮ ⋮ ⋱ ⋮ f 1 ( α m ) f 2 ( α m ) ⋯ f n ( α m ) ] {\displaystyle M={\begin{bmatrix}f_{1}(\alpha _{1})&f_{2}(\alpha _{1})&\cdots &f_{n}(\alpha _{1})\\f_{1}(\alpha _{2})&f_{2}(\alpha _{2})&\cdots &f_{n}(\alpha _{2})\\f_{1}(\alpha _{3})&f_{2}(\alpha _{3})&\cdots &f_{n}(\alpha _{3})\\\vdots &\vdots &\ddots &\vdots \\f_{1}(\alpha _{m})&f_{2}(\alpha _{m})&\cdots &f_{n}(\alpha _{m})\\\end{bmatrix}}} or, more compactly, M i j = f j ( α i ) {\displaystyle M_{ij}=f_{j}(\alpha _{i})} . (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which f j ( α ) = α j − 1 {\displaystyle f_{j}(\alpha )=\alpha ^{j-1}} , and Moore matrices, for which f j ( α ) = α q j − 1 {\displaystyle f_{j}(\alpha )=\alpha ^{q^{j-1}}} .
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose. That is, if the element in the j {\displaystyle j} -th row and k {\displaystyle k} -th column of a Hermitian matrix A {\displaystyle A} is some complex number A j k = x + i y {\displaystyle A_{jk}=x+iy} , then the element in the k {\displaystyle k} -th row and j {\displaystyle j} -th column is its complex conjugate A k j = A j k ¯ = x − i y {\displaystyle A_{kj}={\overline {A_{jk}}}=x-iy} , for every pair of indices j {\displaystyle j} and k {\displaystyle k} . Hermitian matrices can be understood as the complex generalization of symmetric real matrices. Using the notation A T {\displaystyle \textstyle A^{\mathsf {T}}} to mean the transpose of A {\displaystyle A} and an overline to mean the entrywise complex conjugate of a matrix, the Hermitian property is equivalent to the equality A = A T ¯ . {\displaystyle A={\overline {A^{\mathsf {T}}}}.} The conjugate transpose of a matrix is often denoted A H {\displaystyle \textstyle A^{\mathsf {H}}} , in terms of which the Hermitian property can be more concisely expressed as A = A H {\displaystyle \textstyle A=A^{\mathsf {H}}} . Equivalent notations in common use include A H {\displaystyle \textstyle A^{\mathsf {H}}} , A † {\displaystyle \textstyle A^{\dagger }} , and A ∗ {\displaystyle \textstyle A^{*}} , although in quantum mechanics, A ∗ {\displaystyle \textstyle A^{*}} typically means the complex conjugate only, and not the conjugate transpose. Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share with symmetric real matrices the property of always having real eigenvalues.
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an ( m + 1 ) × ( n + 1 ) {\displaystyle (m+1)\times (n+1)} matrix V = V ( x 0 , x 1 , ⋯ , x m ) = ( 1 x 0 x 0 2 … x 0 n 1 x 1 x 1 2 … x 1 n 1 x 2 x 2 2 … x 2 n ⋮ ⋮ ⋮ ⋱ ⋮ 1 x m x m 2 … x m n ) {\displaystyle V=V(x_{0},x_{1},\cdots ,x_{m})={\begin{pmatrix}1&x_{0}&x_{0}^{2}&\dots &x_{0}^{n}\\1&x_{1}&x_{1}^{2}&\dots &x_{1}^{n}\\1&x_{2}&x_{2}^{2}&\dots &x_{2}^{n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{m}&x_{m}^{2}&\dots &x_{m}^{n}\end{pmatrix}}} with entries V i , j = x i j {\displaystyle V_{i,j}=x_{i}^{j}} , the jth power of the number x i {\displaystyle x_{i}} , for all zero-based indices i {\displaystyle i} and j {\displaystyle j} . Some authors define the Vandermonde matrix as the transpose of the above matrix. The determinant of a square Vandermonde matrix (when n = m {\displaystyle n=m} ) is called a Vandermonde determinant or Vandermonde polynomial. Its value is: det ( V ) = ∏ 0 ≤ i < j ≤ n ( x j − x i ) = ( − 1 ) n ( n + 1 ) / 2 ∏ 0 ≤ i < j ≤ n ( x i − x j ) . {\displaystyle \det(V)=\prod _{0\leq i
In linear algebra, a Moore matrix, introduced by E. H. Moore (1896), is a matrix defined over a finite field. When it is a square matrix, its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism applied to its columns (beginning with the zeroth power of the Frobenius automorphism in the first column), so it is an m × n matrix M = [ α 1 α 1 q … α 1 q n − 1 α 2 α 2 q … α 2 q n − 1 α 3 α 3 q … α 3 q n − 1 ⋮ ⋮ ⋱ ⋮ α m α m q … α m q n − 1 ] {\displaystyle M={\begin{bmatrix}\alpha _{1}&\alpha _{1}^{q}&\dots &\alpha _{1}^{q^{n-1}}\\\alpha _{2}&\alpha _{2}^{q}&\dots &\alpha _{2}^{q^{n-1}}\\\alpha _{3}&\alpha _{3}^{q}&\dots &\alpha _{3}^{q^{n-1}}\\\vdots &\vdots &\ddots &\vdots \\\alpha _{m}&\alpha _{m}^{q}&\dots &\alpha _{m}^{q^{n-1}}\\\end{bmatrix}}} or M i , j = α i q j − 1 {\displaystyle M_{i,j}=\alpha _{i}^{q^{j-1}}} for all indices i and j. (Some authors use the transpose of the above matrix.) The Moore determinant of a square Moore matrix (so m = n) can be expressed as: det ( M ) = ∏ c ( c 1 α 1 + ⋯ + c n α n ) , {\displaystyle \det(M)=\prod _{\mathbf {c} }\left(c_{1}\alpha _{1}+\cdots +c_{n}\alpha _{n}\right),} where c runs over a complete set of direction vectors, made specific by having the last non-zero entry equal to 1, i.e., det ( M ) = ∏ 1 ≤ i ≤ n ∏ c 1 , … , c i − 1 ( c 1 α 1 + ⋯ + c i − 1 α i − 1 + α i ) . {\displaystyle \det(M)=\prod _{1\leq i\leq n}\prod _{c_{1},\dots ,c_{i-1}}\left(c_{1}\alpha _{1}+\cdots +c_{i-1}\alpha _{i-1}+\alpha _{i}\right).} In particular the Moore determinant vanishes if and only if the elements in the left hand column are linearly dependent over the finite field of order q. So it is analogous to the Wronskian of several functions. Dickson used the Moore determinant in finding the modular invariants of the general linear group over a finite field.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →The Greenwood District.