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

In summary, a matrix derivative is the derivative of a matrix function, which is a function that takes in a matrix as an input and outputs a matrix. It is calculated using the standard rules of differentiation and has various applications in fields like physics, engineering, economics, and computer science. Not all matrix functions can be differentiated and there are special properties to consider, such as the order of multiplication and the derivative of a matrix transpose.
  • #1
sauravrt
15
0
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]
 
Last edited:
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  • #2
Yes it is.
 

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

1. What is a matrix derivative?

A matrix derivative is the derivative of a matrix function, which is a function that takes in a matrix as an input and outputs a matrix. It is used in calculus and linear algebra to calculate the rate of change of a matrix function at a particular point.

2. How is a matrix derivative calculated?

A matrix derivative is calculated using the standard rules of differentiation, such as the product rule, quotient rule, and chain rule. The resulting derivative is a matrix of the same size as the original matrix function.

3. What are the applications of matrix derivatives?

Matrix derivatives have many applications in various fields such as physics, engineering, economics, and computer science. They are used to solve optimization problems, perform regression analysis, and model complex systems.

4. Can any matrix function be differentiated?

No, not all matrix functions can be differentiated. The function must be continuous and differentiable at the point where the derivative is being calculated. Additionally, some functions may not have a closed-form derivative and may require numerical methods to approximate the derivative.

5. Are there any special properties of matrix derivatives?

Yes, there are a few special properties of matrix derivatives that are different from scalar derivatives. For example, the order of multiplication matters, and the derivative of a matrix transpose is the transpose of the derivative. Additionally, the derivative of a scalar times a matrix is equal to the scalar times the derivative of the matrix.

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