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In linear algebra, an invertible matrix is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their ...
The differential ′ is a linear operator that maps an n × n matrix to a real number. Proof. Using the definition of a ... For an invertible matrix A, ...
) To prove that the backward direction + + is invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix Y {\displaystyle Y} (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix X {\displaystyle X} (in this case A + u v T {\displaystyle A+uv^{\textsf {T}}} ) if ...
This gives a formula for the inverse of A, provided det(A) ≠ 0. In fact, this formula works whenever F is a commutative ring, provided that det(A) is a unit. If det(A) is not a unit, then A is not invertible over the ring (it may be invertible over a larger ring in which some non-unit elements of F may be invertible).
In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury [1] [2] – says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix.
According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function f : R n → R n is the Jacobian matrix of the inverse function. That is, the Jacobian matrix of the inverse function at a point p is
And since P is invertible, we multiply the equation from the right by its inverse, finishing the proof. The set of matrices of the form A − λB, where λ is a complex number, is called a pencil; the term matrix pencil can also refer to the pair (A, B) of matrices. [14]
If each B i is invertible then so is D and D −1 (PAP −1) is equal to the identity plus a nilpotent matrix. But such a matrix is always invertible (if N k = 0 the inverse of 1 − N is 1 + N + N 2 + ... + N k−1) so PAP −1 and A are both invertible. Therefore, many of the spectral properties of A may be deduced by applying the theorem to ...