Antisymmetry of a Matrix

Question

A matrix is antisymmetric, or skew-symmetric, if its transpose equals its negative.

The transpose of a matrix can be obtained by reflecting its elements across the main diagonal. Examples of transpositions can be seen here:

\$\begin{pmatrix}11&12&13\21円&22&23\end{pmatrix}\rightarrow\begin{pmatrix}11&21\12円&22\13円&23\end{pmatrix}\$

\$\begin{pmatrix}11&12&13\21円&22&23\31円&32&33\end{pmatrix}\rightarrow\begin{pmatrix}11&21&31\12円&22&32\13円&23&33\end{pmatrix}\$

This matrix is antisymmetric because it equals its transpose when multiplied by -1:

\$\begin{pmatrix}0&2&-1\\-2&0&0\1円&0&0\end{pmatrix}\$

All antisymmetric matrices exhibit certain characteristics:

• Antisymmetry can only be found on square matrices, because otherwise the matrix and its transpose would be of different dimensions.

• Elements which lie on the main diagonal must equal zero because they do not move and consequently must be their own negatives, and zero is the only number which satisfies \$x=-x\$.

• The sum of two antisymmetric matrices is also antisymmetric.

The Challenge

Given a square, non-empty matrix which contains only integers, check whether it is antisymmetric or not.

Rules

• This is code-golf so the shortest program in bytes wins.

• Input and output can assume whatever forms are most convenient as long as they are self-consistent (including output which is not truthy or falsy, or is truthy for non-antisymmetry and falsy for antisymmetry, etc).

• Assume only valid input will be given.

Test Cases

In:
1 1 1
1 1 1
1 1 1
Out: False
In:
 0 0 1
 0 0 0
-1 0 0
Out: True
In:
0 -2
2 0
Out: True

• TwilightSparkle: Speaking of skew-symmetry... That's totally different though since that one is in the CA sense.
• Luis Mendo: What type of outputs can be used? Any two consistent values? Any truthy and falsy values? Can we choose falsy for antisymmetric and truthy for symmetric?
• Luis Mendo: Will the input ever contain complex numbers? Only contain real numbers? Only integers?
• golf69: @LuisMendo I do believe your first comment is addressed by rule 2, but examples were added anyway. Additionally, only integers will be present (also added). For the record I do want to delete this question but I can't.
• TwilightSparkle: @user That's Do X Without Y, obviously, and is thus deprecated.

Answer 1

APL (Dyalog Unicode), 3 bytes

-≡⍉

Try it online!

This is exactly an APLcart entry on "antisymmetric". Basically it checks if the input's negative - matches ≡ the input's transpose ⍉.

Answer 2

Python 2, 45 bytes

lambda A:A==[[-x for x in R]for R in zip(*A)]

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• xnor: 44 bytes

Answer 3

R, 23 bytes

function(m)!any(m+t(m))

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Checks whether there are any non-zero elements in \$M+M^T\$.

Answer 4

Brachylog, 5 bytes

5 bytes seems to be the right length for this (unless you're Jelly). Actually, this would be three bytes if Brachylog implicitly vectorized predicates like negation.

\ṅm2?

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Explanation

\ Transpose
 ṅm2 Map negation at depth 2
 ? Assert that the result is the same as the input

• Daniel W.: How do you program in this language without using something like windows character map or a special keyboard with 5000 keys?
• DLosc: @DanielW. Good question! I have two methods: copying and pasting from the codepage chart, and using a bookmarklet of Adám's language bar (also available for several other languages).

Answer 5

C (gcc), (削除) 67 (削除ここまで) 64 bytes

-3 thanks to AZTECCO

i,j;f(m,s)int**m;{for(i=j=0;i=i?:s--;)j|=m[s][--i]+m[i][s];m=j;}

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Returns 0 if the matrix is antisymmetric, and a nonzero value otherewise.

Answer 6

Octave, 19 bytes

@(a)isequal(a',-a);

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The semicolon doesn't need to be there, but it outputs the function otherwise, so I'll take the one-byte hit to my score for now.

Explanation

It's pretty straightforward - it checks to see if the matrix of the transpose is equal to the negative matrix

Answer 7

JavaScript (ES6), 42 bytes

Returns false for antisymmetric or true for non-antisymmetric.

m=>m.some((r,y)=>r.some((v,x)=>m[x][y]+v))

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Answer 8

Io, 67 bytes

method(~,~map(i,,円\map(I,V,V+x at(I)at(i)))flatten unique==list(0))

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Explanation

For all a[x][y], it checks whether all a[x][y]+a[y][x]==0.

method(~, // Input x.
 ~ map(i,,円 // Map all x's rows (index i):
 \ map(I,V, // Foreach the rows (index I):
 V+x at(I)at(i) // x[i][I] + x[I][i]
 )
 ) flatten // Flatten the resulting list
 unique // Uniquify the list
 ==list(0) // Does this resulting list *only* contain the item 0?
)

• TwilightSparkle: Hi "new contributor" +1~~~

Answer 9

Pyth, 5 bytes

qC_MM

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Explanation

qC_MM
q : Check if input equals
 C : Transpose of
 _MM : Negated input

Answer 10

MATL, 5 bytes

!_GX=

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Explanation

!_GX=
 // Implicit input on top of stack
! // Replace top stack element with its transpose
 _ // Replace top stack element with its negative
 G // Push input onto stack
 X= // Check for equality

Answer 11

Japt, 5 bytes

eUy®n

Try it

e compare input with :
 Uy columns of input
 ®n with each element negated

Previous version ÕeËËn didn't work, corrected using the ® symbol

Answer 12

Charcoal, 10 bytes

=θEθE豧λκ

Try it online! Link is to verbose version of code. Outputs a Charcoal boolean, i.e. - if the matrix is antisymmetric, nothing if not. Explanation:

 Eθ Map over input matrix rows (should be columns, but it's square)
 Eθ Map over input matrix rows
 §λκ Cell of transpose
 ± Negated
=θ Does matrix equal its negated transpose?

Answer 13

Wolfram Mathematica, (削除) 20 (削除ここまで), 7 bytes

There is a built-in function for this task:

AntisymmetricMatrixQ

But one can simply write a script with less byte counts:

#==-#T&

The T character, as it is displayed in notebooks, stands for transpose. But if you copy this into tio, it won't be recognized because these characters are only supported by Mathematica notebooks.

• M. Stern: I don't think the transpose sign (Unicode: F3C7) should count as one byte?
• polfosol ఠ_ఠ: @M.Stern Maybe I am wrong, but I look at it the same way as ⍉ in the highest voted answer counts as one byte.
• att: @polfosolఠ_ఠ APL has a custom code page. The character for Transpose is , which is 3 bytes. Try it online!

Answer 14

Julia 1.0, 9 bytes

A->A==-A'

A straightforward anonymous function checking the equality.

Try it online!

Answer 15

Husk, 5 bytes

§=T†_

Try it online!

Answer 16

Google Sheets, (削除) 90 (削除ここまで) (削除) 88 (削除ここまで) 41

(Now with lambdas)

Closing parens discounted.

Named Functions (name, args, and formula text all count):

Name	args	Formula
P	a, b	a+b
Q	a	MAP(a,TRANSPOSE(a),P)

Formula:

=SUM(Q(A1:C3))

No, I cannot currently use ADD instead of P as it is not a lambda.

How it Works:

Add the matrix to its transpose. If the resulting matrix is all 0's, then the sum of all elements is 0, which means we the two are equal.

Return 0 if equal, some positive number otherwise.

Answer 17

Haskell, 40 bytes

(==)<*>foldr(zipWith(:).map(0-))z
z=[]:z

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Uses this tip for shorter transpose and the idiom of (==)<*> to check invariance under an operation.

Answer 18

Haskell, (削除) 49 (削除ここまで) 47 bytes

import Data.List 
f x=x==transpose(map(0-)<$>x)

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• saved 2 thanks to @Unrelated String

My first Haskell.
Function tacking a matrix and checking if input is equal to input mapped to (0-value) and transposed

• Unrelated String: -2 with <$>

Answer 19

Scala, 32 bytes

l=>l.transpose==l.map(_.map(-_))

Finally, something that Scala has a builtin for!

The function's pretty straightforward - it compares the transpose of a List[List[Int]](doesn't have to be a List, could be any Iterable) to the negative, found by mapping each list inside l and using - to make it negative.

Try it in Scastie

Answer 20

Ruby 2.7, 40 bytes

->a{a==a.transpose.map{|r|r.map{|c|-c}}}

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• Razetime: I learned a bit of Ruby golfing and fixed it.
• Dingus: Probably this is what you're referring to, but Tips for golfing in Ruby is a good resource if you haven't seen it.
• south: Not saying much but using _1 for either of the map blocks saves two bytes.
• Razetime: it's on tio which used 2.7 or something. guess i'll just add the version number

Answer 21

Pip, 5 bytes

Z_=-_

A function submission; pass a nested list as its argument. Try it online!

Explanation

Z_ The argument, zipped together
 = Equals
 -_ The argument, negated

Answer 22

gorbitsa-ROM, 8 bytes

r1 R A1 B0 T

This is an awful abuse of rule

Input and output can assume whatever forms are most convenient.

If input takes form of "arr[i][j] arr[j][i]", the problem becomes "is sum = 0?".
This code takes pairs of values and outputs their sum if it's not 0

Thus if you provide matrix as previously mentioned pairs, code will return some value for not-anti-symmetric ones and will not return anything for anti-symmetric ones.

r1 R A1 B0 T
r1 #store first number
 R #read second number
 A1 #add first number
 B0 #if sum==0, jump to the beginning
 T #else output the sum

Answer 23

Jelly, 3 bytes

N=Z

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Posting before caird coinheringaahing finds this question.

Answer 24

Java (JDK), (削除) 89 (削除ここまで) (削除) 87 (削除ここまで) 86 bytes

• -2 bytes thanks to Calculuswhiz!

m->{int i=0,j,r=1;for(;++i<m.length;)for(j=0;++j<i;)r=m[i][j]!=-m[j][i]?0:r;return r;}

Try it online!

Returns 0 for false and 1 for true.

• General Grievance: How about int i=0,j then j=0 in the inner loop?
• user: @Calculuswhiz Thanks! I almost never use commas for assignments and totally forgot about them
• General Grievance: Oh, I found something better at 76: m->{int l=m.length,i=l*l;while(--i>=0&&m[i%l][i/l]==-m[i/l][i%l]);return i;}. This returns -1 only if antisymmetric, something bigger otherwise. Your original code might also have needed to start at -1's instead of 0's.
• user: @Calculuswhiz That's really cool, you should post your own answer!

Answer 25

Nekomata + -e, 3 bytes

Ť_=

Attempt This Online!

Ť Transpose
 _ Negate
 = Check equality

Answer 26

Vyxal, 3 bytes

∩N=

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∩ # Transpose
 N # Negate
 = # Equal to original?

Answer 27

Uiua, 4 bytes

≍ ̄⊸⍉

Try it!

Does the ̄ negated ⍉ transposition ≍ match the input (⊸)?

Answer 28

05AB1E, 3 bytes

ø(Q

Try it online or verify all test cases.

Explanation:

ø # Zip/transpose the (implicit) input-matrix; swapping rows/columns
 ( # Negate each value in this transposed matrix
 Q # And check if it's equal to the (implicit) input-matrix
 # (after which the result is output implicitly)

Answer 29

Factor + math.matrices, 19 bytes

[ dup flip mneg = ]

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• dup make a copy of the input
• flip transpose it
• mneg negate it
• = are they equal?

Answer 30

Raku, 26 bytes

{none flat $_ »+«[Z] $_}

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• $_ is the input matrix, in the form of a list of lists of integers.
• [Z] $_ is the transpose of the input matrix.
• »+« adds the input matrix to its transpose.
• flat flattens the matrix into a single list of numbers.
• none converts that list of numbers into a junction which is true if and only if none of the numbers is truthy, that is, nonzero.

The none-junction is returned. It is truthy only if adding the matrix to its transpose results in a matrix containing all zeroes.