Matrix Reference Manual
Matrix Calculus

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Note that the Hermitian transpose is not used because complex conjugates are not analytic.

In the expressions below matrices and vectors A, B, C do not depend on X.

Derivatives of Linear Products

Derivatives of Quadratic Products

Derivatives of Cubic Products

Derivatives of Inverses

Derivative of Trace

Note: matrix dimensions must result in an n*n argument for tr().

Derivative of Determinant

Note: matrix dimensions must result in an n*n argument for det().


If y is a function of x, then dyT/dx is the Jacobian matrix of y with respect to x.

Its determinant, |dyT/dx|, is the Jacobian of y with respect to x and represents the ratio of the hyper-volumes dy and dx. The Jacobian occurs when changing variables in an integration: Integral(f(y)dy)=Integral(f(y(x)) |dyT/dx| dx).

Hessian matrix

If f is a function of x then the symmetric matrix d2f/dx2 = d/dxT(df/dx) is the Hessian matrix of f(x). A value of x for which df/dx = 0 corresponds to a minimum, maximum or saddle point according to whether the Hessian is positive definite, negative definite or indefinite.

The Matrix Reference Manual is written by Mike Brookes, Imperial College, London, UK. Please send any comments or suggestions to