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

Thread starter bernoli123
Start date Apr 30, 2011
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In summary, a skew symmetric matrix is a square matrix where the elements are equal to the negative of their corresponding elements when reflected along the main diagonal. This means that the matrix is symmetric about its main diagonal and all elements on the main diagonal are equal to 0. It differs from a symmetric matrix in that the elements below the main diagonal are equal to the negative of the elements above the main diagonal. Skew symmetric matrices are commonly used in mathematical and scientific fields, as well as in computer graphics and computer vision. To determine if a matrix is skew symmetric, one can check if its transpose is equal to its negative, and the main diagonal will be filled with 0s. Non-square matrices cannot be skew symmetric as they do not satisfy

Apr 30, 2011

bernoli123

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

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Apr 30, 2011

bernoli123

what about thinking of rank-nullity theory
since the dimension of this skew-symmetric matrix=n(n-1)/2
but how to calculate the dim of the AX=0

What is a skew symmetric matrix?

A skew symmetric matrix is a square matrix in which the elements satisfy the property that the element at the ith row and jth column is equal to the negative of the element at the jth row and ith column. In other words, the matrix is symmetric about its main diagonal, with all elements on the main diagonal being equal to 0.

How is a skew symmetric matrix different from a symmetric matrix?

A symmetric matrix is a square matrix in which the elements are equal to their corresponding elements when reflected along the main diagonal. In contrast, a skew symmetric matrix is also symmetric, but with the additional property that the elements on the main diagonal are equal to 0 and the elements below the main diagonal are equal to the negative of the elements above the main diagonal.

What are some common applications of skew symmetric matrices?

Skew symmetric matrices are commonly used in mathematical and scientific fields, such as physics and engineering, to represent physical quantities and equations. They are also used in computer graphics and computer vision to represent rotations and reflections in 3D space.

How can you determine if a matrix is skew symmetric?

A matrix can be determined to be skew symmetric by checking if its transpose is equal to its negative. In other words, if A is a skew symmetric matrix, then A^T = -A. Additionally, the main diagonal of a skew symmetric matrix will be filled with 0s, and the elements above the main diagonal will be equal to the negative of the elements below the main diagonal.

Can a non-square matrix be skew symmetric?

No, a non-square matrix cannot be skew symmetric. The definition of a skew symmetric matrix requires it to be a square matrix with equal number of rows and columns. A non-square matrix cannot satisfy the property of having equal elements on the main diagonal and being symmetric about the main diagonal.