Now that we have the concept down for how to diagonalize a matrix, let's talk about how to diagonalize a symmetric matrix. In order to do that, let's spend this quick lesson talking about what a symmetric matrix is. In general, let's say if for any given matrix P, if P equals P-transpose, then P is symmetric. We're defining symmetric as, given a matrix, if that matrix's transpose is equal to itself, then that matrix is symmetric. How do we tell? Well, if we're given a matrix, let's say 1 3, and 3, negative 3 is a symmetric, well, let's take the first row and turn it into a column. When we have originally learned transpose, it's just taking the rows and turning them into columns which is equivalent to flipping the matrix diagonally about the actual diagonal of the matrix. You can think about it either way. The simpler way for me at least is, just take the rows and input them as columns, is the same as the original matrix. For this case, we have 1, 3 as a row, so we put 1, 3 as a column, we have 3, negative 3 as a row, so we put 3, negative 3 as a column. Now all we have to do is check, is this the same as the original matrix? 1, 3, 1, 3, 3 negative 3, 3, negative 3. Yes, it is symmetric. In general, anything, where you flip it about the diagonal is going to be symmetric. We have a, b, c here, and then let's say x here, x here, y here, y here, and z here, and z here. Something of this form would also be symmetric when you take the row a, x, y, turn into a column a, x, y. As long as each of these entries flipped about the diagonal are equal to each other, we're going to end up with a symmetric matrix. If you're confused about that, again, you can always check by just taking the rows, putting them as columns in the new matrix, and checking if it's the same as the original matrix. It's always the nice, straightforward way to see if something is symmetric. Now if we have a symmetric matrix, we can do some nice things to diagonalize it. We're going to learn that next time.