# Apply Matrix Order of Operations on Scalars

**URL:** <https://discourse.julialang.org/t/apply-matrix-order-of-operations-on-scalars/102499>\
**Category:** General Usage\
**Tags:** symbolics\
**Created:** [August 4, 2023, 6:19pm UTC](https://discourse.julialang.org/t/apply-matrix-order-of-operations-on-scalars/102499 "2023-08-04T18:19:02Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![TicTac](https://avatars.discourse-cdn.com/v4/letter/t/3ab097/32.png) [@TicTac](https://discourse.julialang.org/u/TicTac)\
**Post date:** [August 4, 2023, 6:19pm UTC](https://discourse.julialang.org/t/apply-matrix-order-of-operations-on-scalars/102499/1 "2023-08-04T18:19:02Z")

</div>

Is there a way to make variables defined by Symbolics (Or any symbolic module I suppose) to be treated with the algebraic rules and order of operations for matrices, while still displaying as scalars?

I am using Symbolics.jl to double check some math I am doing, mostly LFTs. The hand calculations use matrices of matrices, so it seemed worth using the same notation in Julia. My general use case is

```julia
using Symbolics, LinearAlgebra

@variables G W1 W2 W3 W4 K1 K2 # All variables are matrices with appropriate dimensions assumed
# Constructing generalized plant P column by column
# 0 and 1 are taking place of zero and identity matrices.  
C1 = [0, 0, 0, 0, 1, 0]
C2 = [0, W2*G, W3*G, W4, 0, G]
C3 = [0, 0, W3, 0, 0, 1]
C4 = [W1, W2*G, W3*G, W4, 0, G]
P = [C1 C2 C3 C4]

# Partitioning
P11 = P[1:4, 1:3]
P12 = P[1:4, 4]
P21 = P[5:6, 1:3]
P22 = P[5:6, 4]
K = [K2*K1 (-K2)]

# Calculating N via LFT
N = P11 + P12 * K * inv(convert.(Float64, I(2)) - P22 * K)*P21

```

I’m checking that what I derived by hand matches `N`, or various steps along the way like `P22 * K`.

This gives me what seems to be the correct result if everything was scalars. But for matrices, some of the order of operations may get flipped. For instance K[1] has it’s elements flipped.

```julia
julia> K[1]
K1*K2 
# Assumes commutative. Would like to be K2*K1.

```

And I get that this is correct and proper for scalars, so I shouldn’t have anything to complain about.

Inversion creates dividing terms, such as

```julia
julia> inv(I(2) - P22 * K)
2×2 Matrix{Num}:
                1 0
 (G*K1*K2) / (true + G*K2) 1 / (true + G*K2)
# Would like division to be (1 + G*K2)^-1 on left.

```

Further down the line, terms get collected in ways which make sense for scalars, but not for my weird use case.

```julia
julia> K2 * G * K2
G*(K2^2)
# Again, assumes commutative to collect terms. Would like to be K2*G*K2.

```

If I re-write the script using Symbolics arrays (`@variables G[1:2, 1:2], W1[1:2,1:2], etc...`), everything appears correct, and the numerical results appear to match. But I’m trying to check some intermediate result which vanishes when I do this.

I imagine there is a Julia-esque way to do this which overloads the operators. I guess this could be as general as I need some scalar type to be use the matrix order of operations, not just Symbolics types.

---

<div class="post-metadata">

**Author:** ![zdenek\_hurak](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/zdenek_hurak/32/53118_2.png) [@zdenek\_hurak](https://discourse.julialang.org/u/zdenek_hurak)\
**Post date:** [August 4, 2023, 7:29pm UTC](https://discourse.julialang.org/t/apply-matrix-order-of-operations-on-scalars/102499/2 "2023-08-04T19:29:02Z")

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I do not have an answer but I can offer a concise reformulation of your question: what you are looking for is capability to do **noncommutative algebra** in Julia.

Recognising the control theoretic motivation in your post, I recall that a package called [NCAlgebra](https://mathweb.ucsd.edu/~ncalg/) offering the needed functionality was developed by William Helton for Mathematica. I do not know if anything similar is available in Julia.