How to compute arccos for a matrix?

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Omar B. 2020-7-4

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评论： Walter Roberson 2020-7-7

I would like to compute the arccos for a matrix. I know when I want to find log(), exp(), and sqrt () for a matrix , we use logm(A), expm(A) and sqrtm(A) where A is a matrix.

I want to find the following:

x=acos(sqrtm(A)\eye(n))

so, Is it correct to compute it like this ? Or we need to use arccosm(sqrtm(A)\eye(n))? Thank you.

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Walter Roberson 2020-7-4

There are no trig matrix functions in MATLAB, except the ones that work element by element.

Omar B. 2020-7-4

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So, is it correct to compute the following?

x=acos(sqrtm(A)\eye(n))

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回答（1 个）

Walter Roberson 2020-7-5

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It depends what you are trying to calculate.

Using the formulas at https://en.wikipedia.org/wiki/Arccos_(trigonometry)#Generalization then

And so we could potentially generalize that for matrices, there might be some meaning to

arccosm = @(z) 1i * logm(z + sqrtm(z^2 - 1))

I am having difficulty thinking of a context in which there could be physical meaning for this.

If we substitute in 1/sqrtm(A) then

1i*logm(sqrtm(A\eye(n) - 1) + sqrtm(A)\eye(m))

But is there a meaning for this??

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John D'Errico 2020-7-6

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f(A) was not your only mistake. If all you were thinking of is:

f_A = acos(1./sqrtm(A))/sqrtm(A);

thus an element-wise acos, then that is correct, as an element-wise acos. But you are asking about a matrix form of the acos, which is something completely different, as much different as sqrt(A) would be compared to sqrtm(A).

Your second mistake was that you wrote 1/sqrtm(A). This is not correct at all in MATLAB, and will generate an error for a matrix A.

I assume that you just wanted to compute the inverse of the elements of sqrtm(A)? Or are you asking to compute the MATRIX inverse of the equare root of the MATRIX A? Again, these are totally different things.

Thus point is, what is the actual formula you are trying to write here? It is impossible to know how to correct this expression when written incorrectly, since it is becoming ambiguous as to what formula you want to compute.

Can we assume that you do need to compute the matrix square root there? Or is that the square root of the elements of A? You used sqrtm, but did you mean to use it? Or did you not appreciate the dirrerence there?

Then you wrote 1/sqrtm(A). This makes the entire question ambiguous, because I cannot know if you intended to form the inverse of the elements or the inverse of the matrix.

And that leads me to question if you are really asking for the acos of the elements or the acos of the matrix.

Omar B. 2020-7-7

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I really appreciate your help. I am really confused about that. I want to evaluat matrix A at the following function

f(x)=arccos(1/sqrt(x))/sqrt(x)

Walter Roberson 2020-7-7

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f1 = @(x) acos(sqrtm(x)^(-1)) * sqrtm(x)^(-1);
f2 = @(x) acos(sqrtm(x)/eye(size(x,1))) / sqrtm(x);
f3 = @(x) acos(1./sqrt(x)) ./ sqrt(x);
f1(A)
f2(A)
f3(A)

Try them all and decide which one is the right solution for you.

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