 So far we've assumed the entries of our matrices are real, but we don't need to. A fundamental extension is to assume the entries are complex numbers. Suppose A is a matrix with complex entries. So something to remember is that if Z is a complex number, the conjugate is a complex number Z bar, where Z Z bar is real and equal to the norm of Z squared. So a natural question to ask, can we find a matrix A bar where A A bar only contains real entries? No. In general, we won't be able to find a conjugate matrix in this sense where the product gives you a matrix with only real entries. Now, while there is no conjugate of a matrix in general, it's still useful to consider the matrix of conjugates. And so we'll use the notation A bar to be the matrix where the entries are the conjugates of the entries. And we claim the following theorem. If A and B are matrices of the appropriate size, then the conjugate of the product is the product of the conjugates. And if it exists, the conjugate of the inverse is the inverse of the conjugate. And you should prove both of these. Now, we can find a conjugate matrix in a special case. When a matrix is a row or column vector, then if A is a row vector, then A times the transpose of the conjugate will be real. And likewise, if A is a column vector, then A times the transpose of the conjugate will be a matrix with real entries. Again, you should prove both of these statements. And this suggests the following definition. The Hermitian conjugate of A, written A-H, is the transpose of the conjugate. If A is equal to A-H, we say that A is an Hermitian matrix. The H and Hermitian are named after a French mathematician by the name of Charles Hermit. So, for example, we can find V-Hermitian and A-Hermitian for two matrices. And remember, this is the transpose of the matrix of conjugates. And remember that finding the conjugate of a complex number involves changing the sign of the pure imaginary part. Since two is a real number, the conjugate of two is just two. The conjugate of three plus I will be three minus I. And the conjugate of four I will be negative four I. And then we transpose the matrix of conjugates. And similarly, the Hermitian of A will be the transpose of the matrix of conjugates. So we find our conjugates and transpose. And what's mathematics without a little proof? Let's prove that provided both sides exist, the Hermitian of a product is the product of the Hermitians in the reverse order. So again, we can always write down one side of inequality, and so we have the Hermitian of a product equal to... Well, definitions are the whole of mathematics. All else is commentary. And since we have the Hermitian, we want to write this as the transpose of the conjugate. But this is the conjugate of a product, and so that will be the product of the conjugates. Now, we know that when we transpose a product, we get the product of the transposes in the reverse order. And definitions are the whole of mathematics. This transpose of a conjugate is exactly how we define the Hermitian. So this conjugate B transpose, that's B Hermitian. And similarly, conjugate A transpose is A Hermitian.