ACO CORP A C CORPORATION of KENTUCKY USA
Founder: TJ Thurmond Morris
ACO-WVO Contract Consultant
American Culture International Relations Ambassadors of Goodwill – Global Tectonic Economics
Administrative Commander – TJ Morris tm ACIR sm ACO Corp Administrative Control USA – ACO-WVO Control (ADCON).
Direction or exercise of authority over subordinate or other
Organizations in respect to administration and support, including organization of Service forces, control of North American organizations for all branches of the military who have served past and present with a friendship of caring in our Community of Practice Skills also known as COPS in USA.
Resources and equipment, personnel management, unit logistics, individual and unit training, readiness,
Mobilization, demobilization, discipline, and other matters not included in the operational missions of the
Subordinate or other organizations. (JP 1-02)
‘via Blog this’
APPENDIX A. A primer on linear matrix algebra
Note: this is a simplified presentation for finite-dimensional real vector spaces. For more general results and rigorous mathematical definitions, refer to mathematical textbooks.
Matrix. A matrix of dimension is a two-dimensional array of real coefficients where is the line index, is the column index. A matrix is usually represented as a table:
A matrix for which is called a square matrix.
Diagonal. The diagonal of a square matrix is the set of coefficients . A matrix is called diagonal if all its non-diagonal coefficients are zero.
Transpose. The transpose of a matrix is a matrix denoted with the coefficients defined by i.e. the coefficients and are swapped, which looks like a symmetry with respect to the diagonal:
Symmetry. A square matrix is symmetric if it is equal to its transpose, i.e. . This is equivalent to having for any and . A property of diagonal matrices is that they are symmetric.
Scalar multiplication. A matrix times a real scalar is defined as the matrix with coefficients .
Matrix sum. The sum of two matrices and is defined as the matrix with coefficients . It is easy to see that the sum and scalar multiplication define a vector space structure on the set of matrices (the sum is associative and its neutral element is the zero matrix, with all coefficients set to zero).
Matrix product. The product between an matrix and a matrix is defined as the matrix with coefficients given by
The product is not defined if the number of columns in is not the same as the number of lines in . The product is not commutative in general. The neutral element of the product is the identity matrix defined as the diagonal matrix with values 1 on the diagonal, and the suitable dimension. If the product can be generalized to matrix times vector by identifying the right-hand term of the product with the column of vector coordinates in a suitable basis; then the multiplication (on the left) of a vector by a matrix can be identified to a linear application from to . Likewise, matrices can be identified with scalars.
Matrix inverse. A square matrix is called invertible if there exist an matrix denoted and called inverse of , such that
Trace. The trace of a square matrix is defined as the scalar which is the sum of the diagonal coefficients.
(A, B, C are assumed to be such that the operations below have a meaning)
The transposition is linear:
Transpose of a product:
Inverse of a product:
Inverse of a transpose:
Associativity of the product:
Diagonal matrices: their products and inverses are diagonal, with coefficients given respectively by the products and inverses of the diagonals of the operands.
Symmetric matrices: the symmetry is conserved by scalar multiplication, sum and inversion, but not by the product (in general).
The trace is linear:
Trace of a transpose:
Trace of a product:
Trace and basis change: , i.e. the trace is an intrinsic property of the linear application represented by .
Positive definite matrices. A symmetric matrix is defined to be positive definite if, for any vector , the scalar unless . Positive definite matrices have real positive eigenvalues, and their positive definiteness is conserved through inversion.