# I am confused using tanh(M)
**URL:** https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247
**Category:** New to Julia
**Created:** [December 13, 2019, 8:02pm UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247 "2019-12-13T20:02:52Z")
**Posts on this page:** 7
**Page:** 1
### Author:  [@floriano](https://discourse.julialang.org/u/floriano)
#### Post date: [December 13, 2019, 8:02pm UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247/1 "2019-12-13T20:02:52Z")
Hello,
I am pritty new to Julia and try to find my first steps. I am a bit confused now.
In Python I had this (simplyfied) piece of code:
```julia
import numpy as np
a = [[1,2], [3,4]]
M = np.array(a)
np.tanh(M)
```
It brings me the expected result:
```julia
array([[0.76159416, 0.96402758],
[0.99505475, 0.9993293]])
```
Now I try my luck with Julia:
```julia
M = [1 2
3 4]
tanh(M)
```
Output:
```julia
2×2 Array{Float64,2}:
-0.0320733 0.472079
0.708118 0.676045
```
My question, what is calculated here?
However if I do:
```julia
tanh.(M)
```
I get the expected result again. I think I understand concept of “.” as the elementwise operator. But I thought tanh() and similar are always elementwise. If I am wrong, what is tanh(M) without “.” calculating?
Thanks
Floriano
---
### Author:  [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)
#### Post date: [December 13, 2019, 8:19pm UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247/2 "2019-12-13T20:19:35Z")
In julia, any operation on a matrix will not be element-wise unless you have a `.`. It turns out that for square matrices, you can take a hyperbolic tangent of a matrix. That’s the result you are seeing when you type `tanh(M)`.
---
### Author:  [@floriano](https://discourse.julialang.org/u/floriano)
#### Post date: [December 13, 2019, 8:20pm UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247/3 "2019-12-13T20:20:56Z")
thank you
---
### Author:  [@chakravala](https://discourse.julialang.org/u/chakravala)
#### Post date: [December 21, 2019, 4:51pm UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247/5 "2019-12-21T16:51:33Z")
```julia
julia> tanh(1+2im)
1.1667362572409197 - 0.24345820118572525im
julia> tanh([1 -2; 2 1])
2×2 Array{Float64,2}:
1.16674 0.243458
-0.243458 1.16674
```
---
### Author:  [@jonasaugust](https://discourse.julialang.org/u/jonasaugust)
#### Post date: [December 21, 2019, 5:34pm UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247/6 "2019-12-21T17:34:29Z")
As Oscar\_Smith said, the component-wise `tanh` and (diagonalizable, square) matrix `tanh` are different in general. The matrix `tanh` is derived from a power series expansion of the scalar tanh, where the scalar powers are replaced with matrix powers. It can be computed by taking the eigenvalue decomposition `U*diagm(evals)*U'` of the matrix, taking the component-wise `tanh` on the eigenvalues, and then reconstructing:
```julia
julia> A = [1 -2; 2 1]
2×2 Array{Int64,2}:
1 -2
2 1
julia> using LinearAlgebra
julia> evals, U = eigen(A)
Eigen{Complex{Float64},Complex{Float64},Array{Complex{Float64},2},Array{Complex{Float64},1}}
eigenvalues:
2-element Array{Complex{Float64},1}:
1.0 - 2.0000000000000004im
1.0 + 2.0000000000000004im
eigenvectors:
2×2 Array{Complex{Float64},2}:
0.0+0.707107im 0.0-0.707107im
-0.707107-0.0im -0.707107+0.0im
julia> U*diagm(evals)*U'-A
2×2 Array{Complex{Float64},2}:
-2.22045e-16+0.0im -4.44089e-16+0.0im
4.44089e-16+0.0im 2.22045e-16+0.0im
julia> U*diagm(tanh.(evals))*U'-tanh(A)
2×2 Array{Complex{Float64},2}:
-4.44089e-16+0.0im 2.22045e-16+0.0im
-2.498e-16+0.0im 0.0+0.0im
julia> tanh.(A)-tanh(A)
2×2 Array{Float64,2}:
-0.405142 -1.20749
1.20749 -0.405142
```
The last line gives an example where the component-wise and matrix versions of `tanh` differ.
---
### Author:  [@GunnarFarneback](https://discourse.julialang.org/u/GunnarFarneback)
#### Post date: [December 21, 2019, 6:14pm UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247/7 "2019-12-21T18:14:47Z")
> [@jonasaugust](#):
>
> It can be computed by taking the eigenvalue decomposition `U*diagm(evals)*U'` of the matrix, taking the component-wise `tanh` on the eigenvalues, and then reconstructing:
Provided that the matrix is diagonalizable, of course.
---
### Author:  [@jonasaugust](https://discourse.julialang.org/u/jonasaugust)
#### Post date: [December 22, 2019, 10:56am UTC](https://discourse.julialang.org/t/i-am-confused-using-tanh-m/32247/8 "2019-12-22T10:56:57Z")
Yes, thanks for reminding us that the power series derivation of `tanh(A)` requires diagonalizability of `A`. But my demo used the decomposition `A=U*diagm(evals)*U'`, so we need to further assume that `A` is unitarily diagonalizable for that to work, i.e., the transpose `U'` is also the inverse of `U`, which occurs when `A` is normal (`A*A' == A'*A`). It’s sufficient that `A` is (real) symmetric for normality to hold.