Civil Rights
Movements, leaders, victories and the continuing fight for equality.
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THE BHP WIRE —
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Explore the people, places, events, achievements, struggles and stories that shaped our journey.
Movements, leaders, victories and the continuing fight for equality.
Innovation, patents, science, technology and world-changing contributions.
Pioneers, champions, Negro Leagues, records, activism and excellence.
Meet the people whose lives, choices and achievements shaped the journey.
Black towns, communities, institutions and places where history happened.
Moments that changed communities, movements, institutions and the nation.
Katherine Johnson’s mathematical calculations helped guide some of America’s most important early space missions while she confronted the racial and gender barriers faced by Black women in twentieth-century America.
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 mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero). Some authors use "totally positive" to include all totally non-negative matrices.
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n-matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1)-submatrices of B. Specifically, for every i, the Laplace expansion along the ith row is the equality det ( B ) = ∑ j = 1 n ( − 1 ) i + j b i , j m i , j , {\displaystyle {\begin{aligned}\det(B)&=\sum _{j=1}^{n}(-1)^{i+j}b_{i,j}m_{i,j},\end{aligned}}} where b i , j {\displaystyle b_{i,j}} is the entry of the ith row and jth column of B, and m i , j {\displaystyle m_{i,j}} is the determinant of the submatrix obtained by removing the ith row and the jth column of B. Similarly, the Laplace expansion along the jth column is the equality det ( B ) = ∑ i = 1 n ( − 1 ) i + j b i , j m i , j . {\displaystyle {\begin{aligned}\det(B)&=\sum _{i=1}^{n}(-1)^{i+j}b_{i,j}m_{i,j}.\end{aligned}}} (Each identity implies the other, since the determinants of both a matrix and its transpose are the same.) The coefficient ( − 1 ) i + j m i , j {\displaystyle (-1)^{i+j}m_{i,j}} of b i , j {\displaystyle b_{i,j}} in the above sum is called the cofactor of b i , j {\displaystyle b_{i,j}} in B. The Laplace expansion is often useful in proofs, as in, for example, allowing recursion on the size of matrices. It is also of didactic interest for its simplicity and as one of several ways to view and compute the determinant. For large matrices, it quickly becomes inefficient to compute when compared to Gaussian elimination.
The nullity theorem is a mathematical theorem about the inverse of a partitioned matrix, which states that the nullity of a block in a matrix equals the nullity of the complementary block in its inverse matrix. Here, the nullity is the dimension of the kernel. The theorem was proven in an abstract setting by Gustafson (1984), and for matrices by (Fiedler & Markham 1986). Partition a matrix and its inverse in four submatrices: [ A B C D ] − 1 = [ E F G H ] . {\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}={\begin{bmatrix}E&F\\G&H\end{bmatrix}}.} The partition on the right-hand side should be the transpose of the partition on the left-hand side, in the sense that if A is an m-by-n block then E should be an n-by-m block. The statement of the nullity theorem is now that the nullities of the blocks on the right equal the nullities of the blocks on the left (Strang & Nguyen 2004): nullity A = nullity H , nullity B = nullity F , nullity C = nullity G , nullity D = nullity E . {\displaystyle {\begin{aligned}\operatorname {nullity} \,A&=\operatorname {nullity} \,H,\\\operatorname {nullity} \,B&=\operatorname {nullity} \,F,\\\operatorname {nullity} \,C&=\operatorname {nullity} \,G,\\\operatorname {nullity} \,D&=\operatorname {nullity} \,E.\end{aligned}}} More generally, if a submatrix is formed from the rows with indices {i1, i2, …, im} and the columns with indices {j1, j2, …, jn}, then the complementary submatrix is formed from the rows with indices {1, 2, …, N} \ {j1, j2, …, jn} and the columns with indices {1, 2, …, N} \ {i1, i2, …, im}, where N is the size of the whole matrix. The nullity theorem states that the nullity of any submatrix equals the nullity of the complementary submatrix of the inverse.
In mathematics, a balanced matrix is a matrix in which every entry is either zero or one that does not contain any square submatrix of odd order having all row sums and all column sums equal to 2. Balanced matrices are studied in linear programming. The importance of balanced matrices comes from the fact that the solution to a linear programming problem is integral if its matrix of coefficients is balanced and its right hand side or its objective vector is an all-one vector. In particular, if one searches for an integral solution to a linear program of this kind, it is not necessary to explicitly solve an integer linear program, but it suffices to find an optimal vertex solution of the linear program itself. As an example, the following matrix is a balanced matrix: [ 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 1 ] {\displaystyle {\begin{bmatrix}1&1&1&1\\1&1&0&0\\1&0&1&0\\1&0&0&1\\\end{bmatrix}}}
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 →Mae Jemison, aboard Space Shuttle Endeavour in 1992.
