Is the inverse of a positive definite matrix positive definite?
Is the inverse of a positive definite matrix positive definite?
The matrix inverse of a positive definite matrix is also positive definite. The definition of positive definiteness is equivalent to the requirement that the determinants associated with all upper-left submatrices are positive.
How do you find the determinant of a block matrix?
det(M)=det(A−BD−1C)det(D). 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).
What is the inverse of a triangular matrix?
Inverse of an upper/lower triangular matrix is another upper/lower triangular matrix. Inverse exists only if none of the diagonal element is zero.
Are inner products always positive?
The inner product is positive semidefinite, or simply positive, if ‖x‖2≥0 always. The inner product is positive definite if it is both positive and definite, in other words if ‖x‖2>0 whenever x≠0. The inner product is negative semidefinite, or simply negative, if ‖x‖2≤0 always.
What is the complement of A and B?
Definition. If A and B are sets, then the relative complement of A in B, also termed the set difference of B and A, is the set of elements in B but not in A.
What is the Schur complement in matrix analysis?
Emilie Virginia Haynsworth was the first to call it the Schur complement. The Schur complement is a key tool in the fields of numerical analysis, statistics, and matrix analysis. The Schur complement arises when performing a block Gaussian elimination on the matrix M.
What is the Schur complement of Gaussian elimination?
The Schur complement arises when performing a block Gaussian elimination on the matrix M. In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows: where Ip denotes a p × p identity matrix.
Who first called it the Schur complement?
Emilie Virginia Haynsworth was the first to call it the Schur complement. The Schur complement is a key tool in the fields of numerical analysis, statistics, and matrix analysis.
Are identity matrices and the Schur complement of a matrix invertible?
If and its Schur complement are invertible, then where are identity matrices and are matrices of zeros. Proposition Let be a block matrix such that its blocks and are square matrices. If and its Schur complement are invertible, then where are identity matrices and are matrices of zeros.
