Sylvester's criterion
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester. Sylvester's criterion states that a "n" × "n" Hermitian matrix "M" is positive-definite if and only if all the following matrices have a positive determinant: In other words, all of the "leading" principal minors must be positive. By using appropriate permutations of rows and columns of "M", it can also be shown that the positivity of "any" nested sequence of "n" principal minors of "M" is equivalent to  M being positive-definite. An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the "leading" principal minors: a Hermitian matrix "M" is positive-semidefinite if and only if all principal minors of "M" are nonnegative. Elementary proof. Suppose formula_2is formula_3 Hermitian matrix formula_4. Let formula_5 be the principal minor matrices, the formula_6 upper left corner matrices. Let's show that if formula_7 is positive definite the principal minors are positive that is formula_8. It is easy to see that formula_9 is positive definite by choosing and noticing formula_11 Equivalently  the eigenvalues of formula_9 are positive. The determinant is the product of eigenvalues and therefore positive formula_13. This ends the part 1. Then the other way, we use induction. The most general form of an formula_14 Hermitian matrix is where formula_2 is an formula_17 Hermitian matrix, formula_18 is a vector and formula_19 is a real constant. Suppose the criterion holds for formula_2 we show that it holds for formula_21. We assume that the all the principal minors of formula_21 are positive. This implies that formula_23, formula_24, and that formula_2 is positive definite by the inductive hypothesis. Denote then Let's  complete the square: formula_33 on formula_34 to obtain The block matrix determinant formula follows from taking the determinant of both sides of the following identity: formula_39 Proof. The proof is only for nonsingular Hermitian matrix with coefficients in formula_40, therefore only for nonsingular real-symmetric matrices. Positive definite or semidefinite matrix: A symmetric matrix "A" whose eigenvalues are positive ("λ" > 0) is called positive definite, and when the eigenvalues are just nonnegative ("λ" ≥ 0), "A" is said to be positive semidefinite. Theorem I: A real-symmetric matrix "A" has nonnegative eigenvalues if and only if "A" can be  factored as "A" = "B""B", and all eigenvalues are positive if and only if "B" is nonsingular. Theorem II (The Cholesky decomposition): The symmetric matrix "A" possesses positive pivots if and only if "A" can be uniquely factored as "A = RR", where "R" is an upper-triangular matrix with positive diagonal entries. This is known as the Cholesky decomposition of "A", and "R" is called the Cholesky factor of "A". Theorem III: Let "A" be the "k" × "k" leading principal submatrix of "A". If "A" has an "LU" factorization "A" = "LU", where "L" is a lower triangular  matrix with a unit diagonal, then det("A") = "u""u" · · · "u", and the "k"-th pivot is "u" = det("A") = "a" for "k" = 1, "u" = det("A")/det("A") for "k" = 2, 3, . . . , "n", where "u" is the ("k", "k")-th entry of "U" for all "k" = 1, 2, . . . , "n". Combining Theorem II with Theorem III yields: Statement I: If the symmetric matrix "A" can be factored as "A=RR" where R is an upper-triangular matrix with positive diagonal entries, then all the pivots of "A" are positive (by Theorem II), therefore all the leading principal minors of "A  are positive (by Theorem III). Statement II: If the nonsingular n" × "n" symmetric matrix "A" can be factored as formula_41, then the QR decomposition (closely related to Gram-Schmidt process) of "B" ("B" = "QR") yields: formula_42, where "Q" is orthogonal matrix and "R" is upper triangular matrix. As "A" is non-singular and formula_43, it follows that all the diagonal entries of "R" are non-zero. Let "r" be the ("j", "j")-th entry of "R" for all "j" = 1, 2, . . . , "n". Then "r" ≠ 0 for all "j" = 1, 2, . . . , "n". Let "F" be a diagonal  matrix, and let "f" be the ("j", "j")-th entry of "F" for all "j" = 1, 2, . . . , "n". For all "j" = 1, 2, . . . , "n", we set "f" = 1 if "r" > 0, and we set "f" = -1 if "r" < 0. Then formula_44, the "n" × "n" identity matrix. Let "S"="FR". Then "S" is an upper-triangular matrix with all diagonal entries being positive. Hence we have formula_45, for some upper-triangular matrix "S" with all diagonal entries being positive. Namely Statement II requires the non-singularity of the symmetric matrix "A". Combining Theorem I with Statement  I and Statement II yields: Statement III: If the real-symmetric matrix "A" is positive definite then "A" possess factorization of the form "A" = "B""B", where "B" is nonsingular (Theorem I), the expression "A" = "B""B" implies that "A" possess factorization of the form "A" = "R""R" where "R" is an upper-triangular matrix with positive diagonal entries (Statement II), therefore all the leading principal minors of "A" are positive (Statement I). In other words, Statement III proves the "only if" part of Sylvester's Criterion for non-singular real-symmetric matrices. Sylvester's Criterion: The real-symmetric matrix  A is positive definite if and only if all the leading principal minors of "A" are positive.