#INVERSE MATRIX MATHEMATICA HOW TO#

The determinant of the inverse of a matrix equals to the reciprocal of the determinant of the original matrix.

Transposing a matrix first and then finding the inverse of the matrix is the same as first calculating the inverse of the matrix and then transposing it.

The inverse of a matrix multiplication is equal to the product of the inverses of the matrices but changing their order of multiplication.

The inverse of the inverse matrix results in the original matrix:.That is, if the matrix is invertible, it only exists one inverse matrix. This returns the same result as Mathematica above, up to floating point precision.The inverse matrix has the following characteristics: The pseudoinverse can be computed in NumPy with np.linalg.pinv.

Note that the singular value decompositions as computed by Mathematica and Python differ in a few signs here and there the SVD is not unique. Since Python is doing floating point computations, not symbolic calculation like Mathematica, the zero in A turns into -3.8e-16. This returns the matrix A, within floating point accuracy. We can verify that the SVD is correct by turning s back into a matrix and multiply the components together. This can save a lot of space if the matrix is large. The NumPy method svd has other efficiency-related options that I won’t go into here. Note that np.linalg.svd returns the transpose of V, not the V in the definition of singular value decomposition.Īlso, the object s is not the diagonal matrix Σ but a vector containing only the diagonal elements, i.e. > u, s, vt = np.linalg.svd(a, full_matrices=True) Next we compute the singular value decomposition in Python (NumPy). We can find the SVD of A with the following Mathematica commands. We’ll give examples below in Mathematica and Python. Where Σ + is formed from Σ by taking the reciprocal of all the non-zero elements, leaving all the zeros alone, and making the matrix the right shape: if Σ is an m by n matrix, then Σ + must be an n by m matrix. Not every matrix has an inverse, but every matrix has a pseudoinverse, even non-square matrices.Ĭomputing the pseudoinverse from the SVD is simple. The (Moore-Penrose) pseudoinverse of a matrix generalizes the notion of an inverse, somewhat like the way SVD generalized diagonalization. A matrix M is unitary if its inverse is its conjugate transpose, i.e. But if the matrix has complex entries, you take the conjugate and transpose each entry. If a matrix has all real components, then the conjugate transpose is just the transpose. The star superscript indicates conjugate transpose. Similarly the columns of U and V are not necessarily eigenvectors but left singular vectors and right singular vectors respectively. The elements along the diagonal of Σ are not necessarily eigenvalues but singular values, which are a generalization of eigenvalues. The matrices U and V are square, but not necessarily of the same dimension. The matrices on either side of Σ are analogous to the matrix P in diagonalization, though now there are two different matrices, and they are not necessarily inverses of each other. Σ is diagonal, though it may not be square.

The matrix Σ in SVD is analogous to D in diagonalization. Singular value decomposition generalizes diagonalization. Singular value decomposition is a way to do something like diagonalization for any matrix, even non-square matrices. Unfortunately not all matrices can be diagonalized. The elements on the diagonal of D are the eigenvalues of A and the columns of P are the corresponding eigenvectors. You could think of P as a change of coordinates that makes the action of A as simple as possible. If a square matrix A is diagonalizable, then there is a matrix P such that The singular value decomposition of a matrix is a sort of change of coordinates that makes the matrix simple, a generalization of diagonalization.

#INVERSE MATRIX MATHEMATICA HOW TO#

This post will explain what the terms above mean, and how to compute them in Python and in Mathematica. The Moore-Penrose pseudoinverse is given by In a nutshell, given the singular decomposition of a matrix A,

Inverse matrix mathematica