Determinant of a block-triangular matrix A block-upper-triangular matrix is a matrix of the form where and are square matrices. PropositionLet be a block-upper-triangular matrix, as defined above. Then, Proof A block-lower-triangular matrix is a matrix of the form where and are square matric...
Determinant of a triangular matrixThe first result concerns the determinant of a triangular matrix. Proposition Let be a triangular matrix (either upper or lower). Then, the determinant of is equal to the product of its diagonal entries: Proof...
<idx><h>Matrix</h><h>diagonal entries of</h></idx> <idx><h>Matrix</h><h>upper-triangular</h></idx> <idx><h>Matrix</h><h>lower-triangular</h></idx> <idx><h>Diagonal</h><h>see Matrix</h></idx> <idx><h>Upper-triangular</h><h>see Matrix</h></idx> <idx>...
So the natural question at this point is: What is the determinant of an upper triangular matrix? Property 6: The determinant of an upper triangular (or diagonal) matrix is equal to the product of the diagonal entries. To prove this property, assume that the given matrix A has been reduced...
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A matrix with a row of zeros has det A = 0. If A is triangular then detA=a11a22...anndetA=a11a22...ann=product of diagnonal entries. If A is singular then det A = 0. If A is invertible then detA≠0detA≠0. The determinant of AB is det A times det B : |AB|=|A||B||...
By finding, for the first time, the matrix elements of the SoV measure explicitly we were able to compute correla- tion functions and wave function overlaps in a simple determinant form. In particular, we show how an overlap between on-shell and off-shell algebraic Bethe states can be ...
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Learn the definition of the determinant. Learn some ways to eyeball a matrix with zero determinant, and how to compute determinants of upper- and lower-triangular matrices. Learn the basic properties of the determinant, and how to apply them. Recipe: compute the determinant using row and c...