What is Skew symmetric: Definition and 20 Discussions

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition

In terms of the entries of the matrix, if



{\textstyle a_{ij}}
denotes the entry in the


{\textstyle i}
-th row and


{\textstyle j}
-th column, then the skew-symmetric condition is equivalent to

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  1. T

    A Reason out the cross product (for the moment): a skew symmetric form

    I am sure you are all familiar with the cross product in 3D space. i cross into j gives k. Cyclic Negative, if reversed, etc. I am sure you are all familiar with the definition as: norm of the first vector, norm of the second, sine of the angle, perpendicular (but direction using right hand...
  2. T

    I Un-skewing a skew symmetric matrix (for want of a better phrase)

    Hello Say I have a column of components v = (x, y, z). I can create a skew symmetric matrix: M = [0, -z, y; z, 0; -x; -y, x, 0] I can also go the other way and convert the skew symmetric matrix into a column of components. Silly question now... I have, in the past, referred to this as...
  3. H

    B Skew symmetric 1 dimension

    Hi, if I have a equation like (just a random eq.) p_dot = S(omega)*p. where p = [x, y, z] is the original states, omega = [p, q, r] and S - skew symmetric. How does the equation appear if i only want a system to have the state z? do I get z_dot = -q*x + p*y. Or is the symmetric not valid so I...
  4. M

    What is the derivative of a skew symmetric matrix?

    Homework Statement Need to prove that the derivative of a rotation matrix is a skew symmetric matrix muktiplied by that rotation matrix. Specifically applying it on the Rodrigues’ formula.Homework EquationsThe Attempt at a Solution I have shown that the cubed of the skew symmetric matrix is...
  5. B

    Prove 3x3 Skew symmetric matrix determinant is equal to zero

    Homework Statement Hi there, I'm happy with the proof that any odd ordered matrix's determinant is equal to zero. However, I am failing to see how it can be done specifically for a 3x3 matrix using only row and column interchanging. Homework Equations I have attached the determinant as an...
  6. J

    Distribution of Non-Gaussian Data: Analysis & Presentation

    Any help would be much appreciated. The problem lies in the non-Gaussian distribution of the sample. If we take the entire data set of total fish catch, the skewness statistic equals 7.463 with a std. error of skewness of 0.39. Accordingly, the Z dist. (7.463/0.39)=19.14. Overall, the...
  7. camilus

    Pfaffian and determinants of skew symmetric matrices

    Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the...
  8. kq6up

    Understanding Skew Symmetric Matrices for Physics - A Helpful Guide

    I am a bit dense when it comes to linear algebra for some reason. I am reviewing math to prepare for a physics grad program, and I am using Mary Boas "Mathematical Methods in the Physical Sciences". She presents the idea of a skew symmetric matrix in the problem set rather than in the text. I...
  9. bhanesh

    What is the minimum rank of a skew symmetric matrix?

    What is minimum possible rank of skew symmetric matrix ?
  10. G

    Solve Invertible Skew Symmetric Matrix: Hints & Tips

    I am asking for some hints to solve this excercise. Given an invertible skew symmetric matrix $A$, then show that there are invertible matrices $ R, R^T$ such that $R^T A R = \begin{pmatrix} 0 & Id \\ -Id & 0 \end{pmatrix}$, meaning that this is a block matrix that has the identity matrix in two...
  11. B

    Proving the Even Rank of Skew Symmetric Matrices: Induction and Other Methods

    how can we prove that the rank of skew symmetric matrix is even i could prove it by induction is there another way
  12. I

    Show that diagonal entries of a skew symmetric matrix are zero.

    I'm pretty inexperienced in proof writing. So not sure if this was valid. If a matrix is skew symmetric then A^T = - A, that is the transpose of A is equal to negative A. This implies that if A = a(i,j), then a(j,i) = -a(i,j). If we're referring to diagonal entries, we can say a(j,j) =...
  13. C

    Basis of skew symmetric matrix

    Homework Statement Let W be a 3x3 matrix where A^t(transpose)=-A. Find a basis for W. Homework Equations Find a basis for W. The Attempt at a Solution I have no idea how to start it.
  14. N

    Proof regarding skew symmetric matrices

    Homework Statement Show that if A is skew symmetric, then Ak is skew symmetric for any positive odd integer k. Homework Equations The Attempt at a Solution Wow, I have no idea how to prove this. I'm guessing there's going to be induction involved. I know that the base case of k...
  15. J

    Skew Symmetric Determinant Proof

    Hi all! I was working on some homework for the linear algebra section of my "Math Methods for Physicists" class and was studying skew symmetric matrices. There was a proof I saw on Wikipedia that proves that the determinant of a skew symmetric matrix is zero if the number of rows is an odd...
  16. F

    Prove the property of skew symmetric matrix

    Homework Statement Hi, I need to prove that if S is a skew-symmetric matrix with NXN dimension and B is any square real-valued matrix, therefore the product of transpose(B), S, and B is also askew symmetric matrix Homework Equations This is what I know so far. 1.Transpose(S) = -S...
  17. T

    Skew symmetric matrix question

    Let A in n x n real matrix. For every x in R^n we have <Ax,x>=0 where < , > is scalar product. prove that A^t=-A (A is skew symmetric matrix)
  18. Q

    Dimension of symmetric and skew symmetric bilinear forms

    Given the vector space consisting of all bilinear forms of a vector space V (let's call it B) it's very easy to prove that B is the direct sum of two subspaces, the subspace of symmetric and the subspace of skew symmetric bilinear forms. How would one go about determining the dimension of these...
  19. I

    Basis of set of skew symmetric nxn matrices

    Hi, I am having trouble with the question above. In general, I have trouble with questions like: What is the basis for all nxn matrices with trace 0? What is the dimension? What is the basis of all upper triangular nxn matrices? What is the dimension? Please help!
  20. J

    Can someone help me about skew symmetric?

    Let A be an nxn skew symmetric mx.(A^T=-A). i) Show that if X is a vector in R^n then (X,AX)=0 ii) Show that 0 is the only possible eigenvalue of A iii)Show that A^2 is symmetric iv)Show that every eigenvalue of A^2 is nonpositive. v)Show that if X is an eigenvector of A^2 , then so is AX...