Professor Dave here, let’s look at complex matrices.
Up until now, we’ve been focusing a lot
on real matrices, that is, matrices where
all of the entries are real numbers.
Now it’s time to consider complex matrices,
which are matrices in which some of the entries
contain complex numbers.
If you need a little review regarding comlex
numbers, what they are and relevant operations,
check out this tutorial from earlier in the series.
Otherwise, let’s dive right in.
We can simply recall that complex numbers
are numbers with a real part, a, and an imaginary
part, bi, where i is the square root of -1,
and as we said, we can find complex numbers
in matrices.
Just the way we can separate complex numbers
into a real component and an imaginary component,
we can separate matrices into a real matrix
A and an imaginary matrix iB.
For example, let’s consider the 2 by 3 matrix
2 + 3i, i, 6 - 4i; 7, 2 - 3i, -i.
We can split this complex matrix up into a
real part plus an imaginary part by keeping
in mind that matrix addition works element
by element.
First for the real part we just take the real
part of each element and put them in a matrix
of the same size, so for the first entry,
we take the 2 from 2 + 3i.
Continuing in this manner will give us 2,
0, 6; 7, 2, 0.
Then we do this again for the imaginary part
of each element, so this time we take the
3i from 2 + 3i.
That will give us i times 3, 1, -4; 0, -3, -1.
We could distribute the i and add the two
matrices together to get what we started with.
Now that we know what complex matrices are,
we must consider the complex conjugates of matrices.
Recall that for ordinary complex numbers,
we learned that the complex conjugate is the
same as switching the sign on the imaginary
term, so a + bi would become a - bi.
Similarly, the conjugate of a complex matrix
A + iB is simply A - iB.
We actually don’t even have to take the
step of splitting up the real and complex
parts, the conjugate of a matrix can be found
immediately by switching the sign on every
i term in the matrix.
So with our previous example, 2 + 3i, i, 6
- 4i; 7, 2 - 3i, -i, the conjugate will be
2 - 3i, -i, 6 + 4i; 7, 2 + 3i, i.
So this is certainly simple enough.
However, when it comes to matrices there is
another operation that is much more important
than taking the conjugate, and this is taking
the “conjugate transpose”.
Finding this is not too difficult, you simply
take the conjugate as we just did, and then
take the transpose of the matrix, which as
we recall, involves making the columns into
rows and the rows into columns.
The reason we want to do this is so that we
can multiply the matrix and the conjugate transpose.
Simply taking the conjugate will not give
us matrices we can multiply if they are not square.
For example our 2 by 3 matrix cannot be multiplied
by its 2 by 3 conjugate, because of the way
matrix multiplication works, which we recall
from a previous tutorial.
But if we take the transpose of the conjugate
we will get a 3 by 2 matrix, and now we can
multiply them with no problem.
We’ve already seen that the conjugate was
2 - 3i, -i, 6 + 4i; 7, 2 + 3i, i, so taking
the transpose we will get 2 - 3i, 7; -i, 2
+ 3i; 6 + 4i, i.
This is our conjugate transpose, also referred
to as the hermitian transpose.
Given a matrix M, the notation for taking
the conjugate transpose is MH, or in some
fields, an M† with this symbol, read as
M dagger.
Now that we understand the conjugate transpose,
let’s look at a special type of matrix called
Hermitian matrices.
These are square matrices whose conjugate
transpose is the same as the original matrix.
For example, consider the matrix 2, 1 - i;
1 + i, 3.
Taking the conjugate first, we get 2, 1 +
i; 1 - i, 3.
Now taking the transpose we end up with 2,
1 - i; 1 + i, 3, which is precisely the matrix
we started with, meaning this matrix is Hermitian.
Finally, recall two things.
First, that real matrices are considered orthogonal
if the columns form an orthonormal basis,
and second, that the inverse of an orthogonal
matrix is the same as its transpose.
For complex matrices, if the columns form
orthonormal vectors, the matrix is said to
be “unitary”, and the conjugate transpose
of the matrix is the same as the inverse of
the matrix.
It’s worth noting that a real unitary matrix
is just an orthogonal matrix.
Let’s consider the matrix U = i over root
2, -1 over root 2; 1 over root 2, -i over root 2.
To check if this is unitary, we can simply
take the conjugate transpose, UH, and see
if UH times U gives the identity matrix.
The conjugate of U will be –i over root
2, -1 over root 2; 1 over root 2, i over root 2.
Then taking the transpose we get UH = -i over
root 2, 1 over root 2; -1 over root 2, i over
root 2.
We will now multiply UH and U and see what
we get.
Doing the matrix multiplication we get -i2
over 2 + 1/2, i over 2 minus i over 2; -i
over 2 plus i over 2, 1/2 minus i2 over 2.
Recall that i squared is equal to negative
one, and then simplifying we get UH times
U equals 1, 0; 0, 1, which is the identity matrix.
Thus this matrix is unitary.
Complex matrices come up a lot in fields like
physics, so it’s important to know how to
work with them.
To make sure we have this under control, let’s
check comprehension.
