- #1
cpp6f
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If I have a matrix valued function A(x) of some scalar x, how do I compute the derivative of the square root of A with respect to x? It seems like it should be simple, but I can't find it anywhere on the internet. Thanks!
Derivative of matrix square root
- Thread starter cpp6f
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In summary, the conversation discusses the computation of the derivative of the square root of a matrix valued function A(x) with respect to the scalar x. The question also clarifies that A(x) is positive definite, and that the principle square root of A is also positive definite. The conversation then mentions the use of the Chain Rule and implicit differentiation to solve the problem.
- #2
tiny-tim
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hi cpp6f! 
how would you define the square root of a matrix?
eg the square root of the 2x2 identity matrix could be itself, or [1,0;0,-1]
how would you define the square root of a matrix?
eg the square root of the 2x2 identity matrix could be itself, or [1,0;0,-1]
- #3
cpp6f
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I forgot to mention that A is positive definitive. So the principle square root of A (the square root that is also positive definitive)
- #4
lavinia
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cpp6f said:
See what you can do with the Chain Rule is what you are saying is A(X) is a one parameter family of positive definite square matrices.
- #5
bpet
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To save you a bit of footwork - implicit differentiation on the equation S(x)*S(x)=A(x) shows that S'(x) is the unique solution to a Sylvester equation.
1. What is the derivative of a matrix square root?
The derivative of a matrix square root is a matrix of the same size as the original matrix, with each element being the derivative of the corresponding element in the square root matrix.
2. How is the derivative of a matrix square root calculated?
The derivative of a matrix square root is calculated using the chain rule, where the derivative of the square root function is multiplied by the derivative of the matrix inside the square root.
3. Why is the derivative of a matrix square root important in mathematics?
The derivative of a matrix square root is important in mathematics because it allows us to find the rate of change of a matrix with respect to its square root. This can be useful in various fields, such as optimization, control theory, and statistics.
4. Can the derivative of a matrix square root be negative?
Yes, the derivative of a matrix square root can be negative. This means that as the value of the matrix increases, the value of its square root decreases.
5. Is the derivative of a matrix square root always defined?
No, the derivative of a matrix square root is not always defined. It is only defined for matrices that have real and positive eigenvalues. If a matrix has complex or negative eigenvalues, the derivative of its square root is not well-defined.
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