Continuous square root function on the space of nxn matrices

In summary, the author has proved the existence of a "square root function" on an open epsilon-ball about the identity matrix "I". This function is such that [f(A)]^2=A\qquad \forall\, A \,\text{ s.t.}\, \|I-A\|<\epsilon within Mn, the space of n*n matrices. However, the author was wondering whether there exists a function f such that f2(A)=A \forall A \in M_n.
  • #1
Mathmos6
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0

Homework Statement


Hi again all,
I've just managed to prove the existence (non-constructively) of a 'square root function' f on some open epsilon-ball about the identity matrix 'I' such that [itex][f(A)]^2=A\qquad \forall\, A \,\text{ s.t.}\, \|I-A\|<\epsilon[/itex] within Mn, the space of n*n matrices (note that's f(A)^2, not f^2(A), so for example the identity function wouldn't work) - I used the inverse function theorem on A^2 to deduce its existence. However, I was wondering whether there exists a function f such that f2(A)=A [itex]\forall A \in M_n[/itex]? Or does there only exist such a function in a finite ball? What about for something like a cube root or a quintuple root function?

I was thinking perhaps some sort of compactness argument might work, but I couldn't reason anything in particular (and that's only if it isn't possible for the whole of [itex]M_n[/itex], otherwise compactness wouldn't work I don't suppose)

Many thanks for any help :)
 
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  • #2
Any thoughts, anyone? Any help at all would be appreciated!
 
  • #3
This is actually a fairly substantial question. Not every matrix has a square root; for instance, [tex]\begin{pmatrix}0&1\\0&0\end{pmatrix}[/tex] does not. But positive semidefinite matrices do; there are several possible constructions. The most "fundamental" one from a functional analysis point of view uses something called the functional calculus, which makes it possible to "apply" many real functions, not just the square root function, to linear operators.

Look at the Wikipedia article "Square root of a matrix" to start. For a more pedagogical treatment, there is lots of good material about spectral theory in finite-dimensional spaces (which is what this is related to) in Paul Halmos's book Finite-dimensional vector spaces.
 
  • #4
ystael said:
This is actually a fairly substantial question. Not every matrix has a square root; for instance, [tex]\begin{pmatrix}0&1\\0&0\end{pmatrix}[/tex] does not. But positive semidefinite matrices do; there are several possible constructions. The most "fundamental" one from a functional analysis point of view uses something called the functional calculus, which makes it possible to "apply" many real functions, not just the square root function, to linear operators.

Look at the Wikipedia article "Square root of a matrix" to start. For a more pedagogical treatment, there is lots of good material about spectral theory in finite-dimensional spaces (which is what this is related to) in Paul Halmos's book Finite-dimensional vector spaces.

Thanks ever so much, how stupid of me not to spot that! Is it true of every order then, that there exists some matrix which isn't the n'th power of any matrix? (I couldn't come up with an example this late at night for the cube root, but I may be being slow ;-))
 

Related to Continuous square root function on the space of nxn matrices

1. What is a continuous square root function on the space of nxn matrices?

A continuous square root function on the space of nxn matrices is a mathematical function that takes a square matrix as its input and returns another square matrix that, when multiplied by itself, results in the original input matrix. This function is continuous, meaning that small changes in the input matrix will result in small changes in the output matrix.

2. How is a continuous square root function on the space of nxn matrices different from a regular square root function?

A regular square root function, such as the one used for finding the square root of a number, only operates on single values. A continuous square root function on the space of nxn matrices, on the other hand, operates on matrices and returns matrices as its output.

3. What is the significance of a continuous square root function on the space of nxn matrices?

This type of function has applications in various fields, such as physics, engineering, and computer science. It is used to solve complex mathematical problems and is essential in many advanced calculations and simulations.

4. How is a continuous square root function on the space of nxn matrices calculated?

The calculation of this function involves finding the eigenvalues and eigenvectors of the input matrix. These values are then used to construct the output matrix, which is the square root of the input matrix. The process can be complex and may require advanced mathematical techniques.

5. Are there any limitations to a continuous square root function on the space of nxn matrices?

Yes, there are limitations. This function can only be applied to square matrices, meaning that the number of rows and columns must be equal. Additionally, not all matrices have a square root, so the function may not be defined for certain matrices.

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