## What is Sylvester Theorem?

Sylvester theorem on spherical harmonics. Sylvester’s criterion, a characterization of positive-definite Hermitian matrices. Sylvester’s inequality about the rank (linear algebra) of the product of two matrices. Sylvester’s closed solution for the Frobenius coin problem when there are only two coins.

## What is Sylvester matrix equation?

In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: Then given matrices A, B, and C, the problem is to find the possible matrices X that obey this equation. All matrices are assumed to have coefficients in the complex numbers.

How do you calculate the leading principal minor?

Then the leading principal minors are D1 = a and D2 = ac − b2. If we want to find all the principal minors, these are given by ∆1 = a and ∆1 = c (of order one) and ∆2 = ac − b2 (of order two).

What is a positive semi definite matrix?

A positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonnegative. A matrix. may be tested to determine if it is positive semidefinite in the Wolfram Language using PositiveSemidefiniteMatrixQ[m].

### How do you find the determinant of a block matrix?

det ( M ) = det ( A − B D − 1 C ) det ( D ) . (the determinant of a block triangular matrix is the product of the determinants of its diagonal blocks). If m=n and if C,D commute then det(M)=det(AD−BC) det ( M ) = det ( A D − B C ) .

### What is a resultant matrix?

Definition. The resultant of two univariate polynomials over a field or over a commutative ring is commonly defined as the determinant of their Sylvester matrix. More precisely, let. and. be nonzero polynomials of degrees d and e respectively.

The principal submatrices of a matrix are the matrix itself and those submatrices obtained from it by repeatedly striking out a row and the column of the same index. The leading principal sub matrices are Lhose obtained by striking out exactly one row and its cOlTesponding column.

What is negative semidefinite matrix?

A negative semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonpositive. A matrix. may be tested to determine if it is negative semidefinite in the Wolfram Language using NegativeSemidefiniteMatrixQ[m].

#### What is Sylvester’s theorem?

Sylvester’s theorem or the Sylvester theorem may refer to any of several theorems named after James Joseph Sylvester : The Sylvester–Gallai theorem, on the existence of a line with only two of n given points. Sylvester’s determinant identity. Sylvester’s matrix theorem, also called Sylvester’s formula, for a matrix function in terms of eigenvalues.

#### What is Sylvester’s triangle problem?

Sylvester’s triangle problem, a particular geometric representation of the sum of three vectors of equal length The Weinstein–Aronszajn identity, stating that det ( I + AB) = det ( I + BA ), for matrices A , B, is sometimes attributed to Sylvester.

What are some interesting mathematical facts about Sylvester?

Sylvester theorem on spherical harmonics. Sylvester’s criterion, a characterization of positive-definite Hermitian matrices. Sylvester’s inequality about the rank (linear algebra) of the product of two matrices. Sylvester’s closed solution for the Frobenius coin problem when there are only two coins.

What is Sylvester’s closed solution to Frobenius problem?

Sylvester’s closed solution for the Frobenius coin problem when there are only two coins. Sylvester’s triangle problem, a particular geometric representation of the sum of three vectors of equal length