In this lecture, we'll talk about properties of skew-symmetric matrices and the hat operator, which we saw in the expressions for angular velocity. First let's define an operation called the matrix transpose, denoted by the superscript T. Let A be an n by m matrix, and let A i, j denote the element in the ith row and jth column of A. The i, jth component of the transpose of A is the j, ith component of A. Loosely speaking, the transpose operation flips the rows columns of the matrix A. Consider the following two by two matrix. The transpose of this matrix swaps the terms in the 2, 1 and 1, 2 positions. Note that the diagonal of the matrix always remains unchanged after taking the transpose. Consider the following rectangular matrix. The transpose of this matrix looks like this. Notice how for rectangular matrices, the dimensions of the transpose is different from the dimensions of the original matrix. In this example, the original matrix is a two by three matrix. The transpose becomes a three by two matrix. Consider the matrix that is the result of the previous problem. The transpose of this matrix is calculated here. Notice that we get back to the original matrix from the last problem. We see from this example that the transpose of A transpose is A itself. Finally, consider the one by three matrix which is simply a row vector. The transpose operation turns this row vector into a column vector. We say that a matrix A is symmetric is A transpose = A. We say that a matrix A is skew-symmetric if A transpose = -A. Let's consider specifically 3x3 skew-symmetric matrices. Consider an arbitrary 3x3 matrix A. Using the definition of skew-symmetric, this matrix is skew-symmetric if the following expression is satisfied. Matching up the components of the two matrices on either side of the expression, we get six constraints that must be satisfied for a to be skew symmetric. We see that the first three constraints say that the components A11, A22 and A33 must equal the negative of themselves. The only scalar that satisfies this criteria is 0. We see that a skew-symmetric matrix must have all 0s on its diagonal. The last three constraints relate the remaining six components of the matrix. Substituting these constraints into the matrix gives us the following general expression for a 3x3 skew-symmetric matrix. In particular, notice that because of the constraints for skew symmetry, this matrix only has three independent parameters. As a result, we can concisely represent any skew symmetric 3x3 matrix as a 3x1 vector. The hat operator allows us to switch between these two representations. Exquisitely, A Hat or A is a three by one vector, it's a three by three skew-symmetric matrix defined by the three components of the vector A. For example, consider the vector, omega = 1, 2, 3. The corresponding skew-symmetric matrix, omega hat is shown here. The hat operator is also used to denote the cross product between two vectors. That is u cross v can be written as u hat times v. This means that the cross product of u and v = to the skew symmetric matrix corresponding to u x v. For this reason, the skew symmetric matrix corresponding to u hat is sometimes denoted as u cross. In lecture, we define the following two angular velocity vectors. Will these angular velocity vectors always exist? Recall that we proved that R transpose R dot, and R dot R transpose are both skew symmetric matrices. From the properties we derived earlier, we then note that we are guaranteed to be able to find angular velocity vectors, omega b and omega s, that satisfy the given definitions of angular velocity.