How Do You Calculate the Derivative of a Matrix with Respect to a Scalar?

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SUMMARY

The discussion focuses on calculating the derivative of a Hermitian matrix with respect to a scalar parameter, specifically \(\theta\). The correct expression for the derivative is confirmed as \(\frac{\partial \mathbf{y^H R^{-1} y}}{\partial \theta} = \mathbf{y^H} \frac{\partial \mathbf{R^{-1}}}{\partial \theta} \mathbf{y}\). Here, \(\mathbf{y}\) is defined as an \(N \times 1\) vector independent of \(\theta\), and \(\mathbf{R}\) is an \(N \times N\) Hermitian matrix dependent on \(\theta\). This conclusion establishes a clear method for performing matrix derivatives in this context.

PREREQUISITES
  • Understanding of matrix calculus, particularly derivatives of matrix functions.
  • Familiarity with Hermitian matrices and their properties.
  • Knowledge of scalar parameters in the context of multivariable calculus.
  • Proficiency in linear algebra, specifically vector and matrix operations.
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  • Study the derivation of the inverse of a matrix, particularly \(\frac{\partial \mathbf{R^{-1}}}{\partial \theta}\).
  • Explore applications of matrix derivatives in optimization problems.
  • Learn about the implications of matrix derivatives in machine learning algorithms.
  • Investigate the use of symbolic computation tools like SymPy for matrix calculus.
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Mathematicians, data scientists, and engineers involved in optimization, machine learning, or any field requiring advanced matrix calculus techniques.

sauravrt
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I'm trying to do a matrix derivative.
[tex]\theta , \text{a scalar, is my parameter for derivative}[/tex]
[tex]\mathbf{y}[/tex] is an Nx1 vector independent of [tex]\theta[/tex]

[tex]\mathbf{R}[/tex]
is an NxN Hermitian matrix a function of [tex]\theta[/tex]

Is the following correct?
[tex] \frac{\partial \mathbf{y^H R^{-1} y}}{\partial \theta} = \mathbf{y^H} \frac{\partial \mathbf{R^{-1}}}{\partial \theta} \mathbf{y}[/tex]
 
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Yes it is.
 

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