What is the dimension of Nxn symmetric matrices?

Let A denote the space of symmetric (n×n) matrices over the field K, and B the space of skew-symmetric (n×n) matrices over the field K. Then dim(A)=n(n+1)/2 and dim(B)=n(n−1)/2.

What is the dimension of the set of all skew-symmetric matrices?

All skew-symmetric (anti-symmetric) matrices (AT = −A). All matrices whose nullspace contains the vector (2, 1, −1). Again, since no matrix here is a linear combination of the others, this is a basis and the dimension of the space of symmetric matrices is 6.

Is skew symmetric matrix vector space?

Yes. The set of skew-symmetric matrices forms a vector space of dimension , with respect to matrix addition and scalar multiplication, because every skew-symmetric matrix is determined by that many numbers.

Do skew-symmetric matrices form a subspace?

Subspace of Skew-Symmetric Matrices and Its Dimension Let V be the vector space of all 2×2 matrices. Let W be a subset of V consisting of all 2×2 skew-symmetric matrices. (Recall that a matrix A is skew-symmetric if AT=−A.) (a) Prove that the subset W is a subspace of V.

What is the dimension of the vector space of all 2×2 matrices?

The vector space of 2×2 matrices under addition over a field F is 4 dimensional. It’s span{(1000),(0100),(0010),(0001)}. These are clearly independent under addition.

What is the dimension of skew Hermitian matrix?

The answer is ½n(n-1). The diagonal elements are all 0, and you can define the ½n(n-1)elements above the diagonal arbitrarily (the elements below the diagonal will be the opposites of the corresponding elements above).

How do you find symmetric and skew-symmetric matrix?

We can obtain the symmetric matrix by adding the matrix and its transpose and dividing it with 2. Similarly, the skew symmetric matrix can be obtained by subtracting the transpose of the matrix from the matrix and diving it with 2. Then we will get the symmetric and skew-symmetric parts of the matrix.

What is the dimension for the vector space consisting of the set of all nxn upper triangular matrices?

In general, an n×n matrix has n(n−1)/2 off-diagonal coefficients and n diagonal coefficients. Thus the dimension of the subalgebra of upper triangular matrices is equal to n(n−1)/2+n=n(n+1)/2.

Is the set of all symmetric matrices a subspace?

Solution. The symmetric matrices form a subspace. If a, b ∈ F, and A, B are symmetric n × n matrices, then aA + bB is symmetric since the transpose obeys the rule (aA + bB)t = aAt + bBt, which gives aA + bB when A and B are symmetric.

Is the set of all 2×2 matrices a vector space?

Example 2 The set V of 2×2 matrices is a vector space using the matrix addition and matrix scalar multiplication. Since the multiplication of a scalar and a 2 × 2 matrix is still a 2 × 2 matrix axiom 6 holds. Prove in a similar way that all the other axioms hold, therefore the set of 2 × 2 matrices is a vector space.