What is the minimum rank of a skew symmetric matrix?

In summary, a skew symmetric matrix is a type of square matrix where the elements below the main diagonal are the negative of the elements above the main diagonal. Its rank is always even and is equal to the number of non-zero rows or columns divided by 2, as it is always equal to the number of non-zero eigenvalues. A skew symmetric matrix cannot have a rank of 0, as its main diagonal must contain all 0s and at least one other row or column must also contain all 0s. The rank of a skew symmetric matrix is significant in determining its properties and behavior in mathematical operations, as well as solving systems of linear equations and determining its null space dimension. Finally, a skew symmetric matrix cannot have a rank
  • #1
bhanesh
9
0
What is minimum possible rank of skew symmetric matrix ?
 
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  • #2
Look at the zero matrix.
 
  • #3
But how can we say that zero matrix is skew symmetric matrix
 
  • #4
If 0 denotes the zero matrix, then 0T + 0 = 0. So this matrix is skew-symmetric.
 
  • #5
a little more surprising question might be what is the maximum rank, say of a 3by3 skew symmetric matrix?
 
  • #6
Determinant of skew symmetric matrix of odd order is always zero. So for skew symmetric matrix its rank will be always even in number. ..
 

FAQ: What is the minimum rank of a skew symmetric matrix?

What is a skew symmetric matrix?

A skew symmetric matrix is a type of square matrix where the elements below the main diagonal are the negative of the elements above the main diagonal. In other words, the matrix is equal to its negative transpose.

How is the rank of a skew symmetric matrix determined?

The rank of a skew symmetric matrix is always even, and is equal to the number of non-zero rows or columns divided by 2. This is because the rank of a skew symmetric matrix is always equal to the number of non-zero eigenvalues.

Can a skew symmetric matrix have a rank of 0?

No, a skew symmetric matrix must have a rank of at least 2. This is because the main diagonal must contain all 0s, and at least one other row or column must also contain all 0s.

What is the significance of the rank of a skew symmetric matrix?

The rank of a skew symmetric matrix is important in determining the properties and behavior of the matrix in various mathematical operations. It can also be used to solve systems of linear equations and determine the dimension of the matrix's null space.

Can a skew symmetric matrix have a rank greater than its size?

No, a skew symmetric matrix cannot have a rank greater than its size. This is because the maximum possible rank for a square matrix is equal to its size. However, the rank can be less than the size of the matrix if it contains a significant number of zero elements.

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