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Matrix Inequalities
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Sylvester's Criterion
A symmetric matrix is positive definite if and only if all leading principal minors of are positive. Example: The matrix is positive definite because all its principal minors (2, 3) are positive.
Fan Inequality
For Hermitian matrices and with eigenvalues and respectively, . Example: If and , then .
Triangle Inequality for Matrices
For any matrices and of the same size, . Example: If and , then should be less than or equal to for any chosen matrix norm.
Schur Product Theorem
For any two positive semidefinite matrices and , their Schur (or Hadamard) product is also positive semidefinite. Example: Let and be positive semidefinite matrices; their Schur product will have non-negative eigenvalues.
Positive Semi-Definite Matrix Sum
If and are positive semidefinite matrices, then is also positive semidefinite. Example: Given that and are positive semidefinite, their sum is also positive semidefinite.
Oppenheim's Inequality
For positive semidefinite matrices and such that is invertible, . Example: Let and then .
Lyapunov's Inequality for Matrices
Given any such that , for matrices and we have . Example: If and are matrices and (Frobenius norm), then .
Cauchy-Schwarz Inequality
For any vectors and in an inner product space, . Example: For and , which simplifies to .
Frobenius Norm Inequality
For any matrices and , . Example: If and are matrices, then the Frobenius norm of their product should not exceed the product of their Frobenius norms.
von Neumann's Trace Inequality
For any matrices and of the same size with singular values and respectively, . Example: If and , then .
Trace Inequality
If is a square matrix and is a positive semidefinite matrix, then . Example: For and being any positive semidefinite matrix, the trace of their product will be non-negative.
Hadamard's Inequality
For any matrix with rows , the determinant satisfies . Example: For matrix , .
Minkowski's Inequality for Matrices
For matrices and under any unitarily invariant norm, . Example: If and are matrices and using the Frobenius norm, then , which holds as an equality if and only if and are positive multiples of each other.
Weyl's Inequality
For Hermitian matrices and , and their sum , the eigenvalues satisfy for all . Example: If and are Hermitian matrices with eigenvalues and respectively, then .
Matrix Holder Inequality
For any matrices and , and with , we have . Example: For (Frobenius norms), for any .
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