Derivative of determinant with respect to matrix I'm studying about EM-algorithm and on one point in my reference the author is taking a derivative of a function with respect to a matrix. It is therefore sufficient to verify the polynomial identity on the dense subset where the eigenvalues of A are distinct and nonzero. Feb 8, 2025 · The derivative of functions expressed in determinant form is important for matrix calculus and is widely used in linear algebra, optimization, and economics. Oct 26, 1998 · Jacobi's Formula for the Derivative of a Determinant Jacobi’s formula is d det(B) = Trace( Adj(B) dB ) in which Adj(B) is the Adjugate of the square matrix B and dB is its differential. By comparing this equation with the total derivative of $J$ with respect to time, you can deduce that $∂J (ξ,t)/∂t = J\nabla\cdot\mathbf {v}$. Conside the following symmetric matrix: Jan 23, 2021 · I want to compute the derivative of the determinant of a matrix. The formula allows us to differentiate determinants involving matrix-valued functions, where the trace and inverse matrices play an essential role. Aug 5, 2025 · What is the Derivative of a Determinant? The derivative of a determinant is an expression that captures how a small change in the elements of a matrix affects its determinant. Here, the second line defines a perfectly good linear operator for the derivative(A−1)′, but if we want we can rewritethisasanexplicitJacobianmatrixbyusingKroneckerproductsactingonthe“vectorized” matricesaswe didinSec. 3: vec d(A−1) = vec −A−1(dA)A−1 Aug 21, 2015 · From this stage, the determinant needs to be calculated. Finally we consider a problem of differentiating matrix determinant, trace and inverse. Oct 27, 2020 · Or perhaps you would prefer to have the partial derivative of your function with respect to a particular entry of the vector $\theta$; this would naturally be presented as a matrix. It highlights in particular the remarkable contribution of a 1948 paper by Dwyer and Macphail which derives the linear and adjoint sensitivities of a matrix product, inverse and determinant, and a number of related results motivated by In vector calculus, the Jacobian matrix (/ dʒəˈkoʊbiən /, [1][2][3] / dʒɪ -, jɪ -/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. Below are the main cases with practical examples: Aug 17, 2015 · Proof for the derivative of the determinant of a matrix [closed] Ask Question Asked 10 years, 2 months ago Modified 5 years, 11 months ago derivative of a determinant of a matrix with respect to an element that appears many times in the matrix Ask Question Asked 12 years, 9 months ago Modified 12 years, 9 months ago Jan 10, 2025 · What are Determinants? Corresponding to every square matrix \ (A,\) there exists a number called the determinant of the matrix. That Suggestions: Your suggestion for additional content or elaboration of some topics is most welcome acookbook@2302. This seems to be relatively straightforward for the first derivative using e. Feb 17, 2015 · How to calculate the derivative with respect to $X$ of: $$ \log \mathrm {det}\, X $$ here $X$ is a positive definite matrix, and det is the determinant of a matrix. We also introduce a new set of Kronecker tensor products of matrices. From the foregoing expressions for directional derivative, we derive a relationship between the gradient with respect to matrix X and the derivative with respect to real variable t : 1 Introduction As explained in detail in [1], there unfortunately exists multiple competing notations concerning the layout of matrix derivatives. Some sources, for example [2] (from which I use a lot of identities), even use a mixed layout (according to [1, Notes Mar 19, 2019 · There, it has the more general form d det (A)=tr (adj (A)dA), where adj (A) is the adjugate of A (the transpose of the cofactor matrix, so that adj (A) = det (A)A−1, if A is invertible). Let Φ(t) be an n × n matrix depending on a parameter t. Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, di erentiate a matrix. We consider in this document : derivative of f with respect to (wrt. We can be more precise by writing the Taylor approximation as Aug 11, 2023 · Here, the left-hand side represents the material derivative of the integral of $f$ over $Ω (t)$, and the right-hand side involves the change of $f$ with respect to time in the material coordinates. Please respect the copyright. May 10, 2020 · Hence the second formula is the correct one for a symmetric matrix. e. Tensor derivative (continuum mechanics) The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. vector is a special case ivative appears in many applications, especially on second o To gain understanding of matrix derivative, we first review scalar derivative and vector derivative Gradient Matrix Example #3: Frobenius Norm Squared There are several possible extensions of Euclidean norms to matrices, of which the Frobenius norm is the most useful. 1 This has the advantage of better agreement of matrix products with composition schemes such as the chain rule. , Jacobi's formula. Jun 14, 2025 · The derivative of determinant is computed using Jacobi's formula, which involves the trace of the product of the adjugate of a matrix and the derivative of the matrix. This greatly simplifies operations Sep 16, 2017 · Derivative of matrix determinant wrt to matrix element Ask Question Asked 8 years, 1 month ago Modified 8 years, 1 month ago 5 Matrix Di erentiation In the following discussion I will di erentiate matrix quantities with respect to the elements of the referenced matrices. g. This formula will be derived and then applied to the rôle of the Wronskian in the solution of linear differential equations, Aug 13, 2022 · The formula holds also for the derivative with respect to a vector, as it gives you the differential of the function $A\mapsto\det A$ along the direction $\dot {A}$. This is a large class of An important family of derivatives with respect to a matrix involves functions of the determinant of a matrix, for example y X or y AX . A few of the books I've seen performing this derivative seem to skip the general formulation of how to form this derivative explicitly for a matrix function argument. ) matrix where the derivative of f wrt. Aug 8, 2014 · I'm working on an expression for the nth derivative of a (symmetric) matrix, i. All formulae require of course the matrix to be non-singular. Both sides of the Jacobi formula are polynomials in the matrix coefficients of A and A'. dk. What's reputation and how do I get it? Instead, you can save this post to reference later. Jan 27, 2023 · Lecture 5 Part 1: Derivative of Matrix Determinant and Inverse Description: The first ~6 minutes are on the topic Norms and Derivatives: Why a norm of the input and output are needed to define a derivative. Jan 5, 2021 · Derivatives with respect to a real matrix If X is p # q and Y is m # n, then dY: = dY / dXdX: where the derivative dY / dX is a large mn # pq matrix. Could someone explain how does one take the derivative of a May 9, 2022 · Did you know that there is a mathematical formula that simplifies finding the derivative of a determinant? You can compute the derivative of a determinant of an n x n matrix by using the sum of n other determinants. The notes can be freely copied for personal and educational purposes, but other uses must be agreed to by the author. Matrix Derivative The derivative of a matrix with respect to either a scalar or vector variable involves calculating the derivative of each element within the matrix, similar to the process used for functions. Upvoting indicates when questions and answers are useful. The first formula is correct for a non symmetric matrix. And, the matrix A is the Jacobian of f at x^ ; that is, it is the first derivative of y with respect to x , evaluated at x^ : ∂yi ∣ A ij = ∂xj ∣ x^ The Jacobian A implicitly depends on the point x^ where we evaluate the derivative. From the foregoing expressions for directional derivative, we derive a relationship between gradient with respect to matrix X and derivative with respect to real variable t : This paper presents a set of rules for matrix differentiation with respect to a vector of parameters, using the flattered representation of derivatives, i. It builds matrix calculus from scratch. If Φ is a dif-ferentiable function of t — that is, each of its components is differentiable with respect to t — then so is det Φ(t), since we know that the determi-nant is a polynomial in the components of Φ. A few proofs, but mostly just references. Only prerequisites Here y = f(x) and y^ = f( x^ ) . This paper collects together a number of matrix derivative results which are very useful in forward and reverse mode algorithmic di erentiation (AD). It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. in form of a matrix. Two of these derivatives may be related to the matrix derivatives of the determinant and the negative trace. Derivative of a Matrix Determinant with Respect to Itself The derivative of the determinant of a matrix with respect to itself is: ¶det[A] = det[A]A T. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Added Let's explain the subtlety with one example that should clarify the matter. This can cause a lot of di culty when consulting several sources, since di erent sources might use di erent conventions. Suppose that we have a matrix Y whose Matrix Calculus[3] is a very useful tool in many engineering prob-lems. Jun 14, 2025 · Dive into the world of linear algebra and vector calculus with our comprehensive guide on the derivative of determinant. The use of these derivative formulas is restricted to "nonderogatory" matrices. In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. [1] Determinant is a multilinear function of the columns (or rows), which means if you fix n-1 of the columns and view it as a function of the n'th column it's linear. However, using ma-trix calculus, the derivation process is more compact. This document is adapted from the notes of a course the author recently attends. Matrix calculus [2], [7] is used to derive formulas for the derivatives of the coefficients of the characteristic polynomial with respect to any matrix of physical parameters. \\begin{equation}\\frac{\\partial^{n} \\det(A)}{\\partial A^{n}}\\end{equation Feb 22, 2014 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Basic rules of matrix calculus are nothing more than ordinary calculus rules covered in undergraduate courses. Both the matrix and (if ¶Aij = dikdjl ¶kl F. Many authors, notably in statistics and economics, define the derivatives as the transposes of those given above. Although no new concept is required to carry out such operations, the element-by-element calculations involve cumbersome manipulations and, thus, it is useful to derive the necessary results and have them readily available 2. The determinant is defined as a function \ (f:M→D,\) where \ (M\) is the set of all square matrices and \ (D\) is a set of values of the determinant of that matrix. Apr 7, 2020 · Readers of this thread may be interested in the "Matrix Cookbook" by Petersen & Pedersen, which has explicit formulas for tons of things like the derivatives of the determinant, eigenvalues, and inverse of a matrix. From that that you can show the derivative of det (M) with respect to entry Mij is the determinant of the matrix you get from M if you put 1 in position (i,j), and otherwise 0 out all other the entries in row i and in column j. Dec 12, 2021 · Derivative of the determinant of the metric Ask Question Asked 3 years, 10 months ago Modified 3 years, 10 months ago A scalar function's derivative with respect to a matrix is a matrix of the same dimension with the partial derivative with respect to each element of the matrix in its corresponding (same) position. It is commonly represented as d/dx (det (A)) for a matrix A. dsebz3 xetl v2fhs1 xutia tmhnnc mjp 1d0mv ggcrlu yygvg 5bwn