Recall that the * conjugate* of a complex number
is . The conjugate of is denoted or .

In this section, I'll use for complex conjugation of numbers of matrices. I
want to use to denote
an operation on matrices, the * conjugate
transpose*.

Thus,

Complex conjugation satisfies the following properties:

(a) If , then if and only if z is a real number.

(b) If , then

(c) If , then

The proofs are easy; just write out the complex numbers (e.g. and ) and compute.

The * conjugate* of a matrix A is the matrix obtained by conjugating each
element: That is,

You can check that if A and B are matrices and , then

You can prove these results by looking at individual elements of the matrices and using the properties of conjugation of numbers given above.

* Definition.* If A is a complex matrix, is the conjugate transpose of A:

Note that the conjugation and transposition can be done in either order: That is, . To see this, consider the element of the matrices:

* Example.* If

Since the complex conjugate of a real number is the real number, if B is a real matrix, then .

* Remark.* Most people call the * adjoint* of A ---
though, unfortunately, the word "adjoint" has already been
used for the transpose of the matrix of cofactors in the determinant
formula for . (Sometimes people
try to get around this by using the term "classical
adjoint" to refer to the transpose of the matrix of cofactors.)
In modern mathematics, the word "adjoint" refers to a
property of that I'll prove below.
This property generalizes to other things which you might see in more
advanced courses.

The operation
is sometimes called the * Hermitian* --- but this
has always sounded ugly to me, so I won't use this terminology.

Since this is an introduction to linear algebra, I'll usually refer
to as the * conjugate
transpose*, which at least has the virtue of saying what the
thing is.

* Proposition.* Let U and V be complex
matrices, and let .

(a) .

(b) .

(c) .

(d) If , their dot product is given by

* Proof.* I'll prove (a), (c), and (d).

For (a), I use the fact noted above that and can be done in either order, along with the facts that

I have

This proves (a).

For (c), I have

For (d), recall that the dot product of complex vectors and is

Notice that you take the complex conjugates of the components of v before multiplying!

This can be expressed as the matrix multiplication

* Example.* In this example, use the complex
dot product.

(a) Compute .

(b) Find .

(c) Find a nonzero vector which is orthogonal to .

(a)

It's a common notational abuse to write the number " " instead of writing it as a matrix " ".

(b)

Hence, .

The following formula is evident from this example:

This extends in the obvious way to vectors in .

(c) I need

In matrix form, this is

Note that the vector was conjugated and transposed.

Doing the matrix multiplication,

I can get a solution by switching the numbers and and negating one of them: .

There are two points about the equation which might be confusing. First,
why is it necessary to conjugate *and* transpose v? The reason
for the conjugation goes back to the need for inner products to be
positive definite (so is a nonnegative
real number).

The reason for the transpose is that I'm using the convention that
vectors are *column vectors*. So if u and v are n-dimensional
column vectors and I want the product to be a number --- i.e. a matrix --- I have to multiply an
n-dimensional *row vector* ( ) and an n-dimensional column
vector ( ). To get the row
vector, I have to transpose the column vector.

Finally, why do u and v switch places in going from the left side to the right side? The reason you write instead of is because inner products are defined to be linear in the first variable. If you use you get a product which is linear in the second variable.

Of course, none of this makes any difference if you're dealing with real numbers. So if x and y are vectors in , you can write

* Definition.* A complex matrix U is * unitary* if .

Notice that if U happens to be a real matrix, , and the equation says --- that is, U is orthogonal. In other
words, *unitary* is the complex analog of *orthogonal*.

By the same kind of argument I gave for orthogonal matrices, implies --- that is, is .

* Proposition.* Let U be a unitary matrix.

(a) U preserves inner products: . Consequently, it also preserves lengths: .

(b) An eigenvalue of U must have length 1.

(c) The columns of a unitary matrix form an orthonormal set.

* Proof.* (a)

Since U preserves inner products, it also preserves lengths of vectors, and the angles between them. For example,

(b) Suppose x is an eigenvector corresponding to the eigenvalue of U. Then , so

But U preserves lengths, so , and hence .

(c) Suppose

Then means

Here is the complex conjugate of the column , transposed to make it a row vector. If you look at the dot products of the rows of and the columns of U, and note that the result is I, you see that the equation above exactly expresses the fact that the columns of U are orthonormal.

For example, take the first row . Its product with the columns , , and so on give the first row of the identity matrix, so

This says that has length 1 and is perpendicular to the other columns. Similar statements hold for , ..., .

* Example.* Find c and d so that the following
matrix is unitary:

This gives

I may take and . Then

So I need to divide each of a and b by to get a unit vector. Thus,

* Proposition.* (*
Adjointness*) let and let . Then

* Proof.*

* Remark.* If is any inner product on a vector
space V and is a linear
transformation, the * adjoint* of T is the linear transformation which
satisfies

(This definition assumes that * there is* such a
transformation.) This explains why, in the special case of the
complex inner product, the matrix is called the *
adjoint*. It also explains the term *
self-adjoint* in the next definition.

* Corollary.* (*
Adjointness*) let and let . Then

* Proof.* This follows from adjointness in the
complex case, because for a real matrix.

* Definition.* An complex matrix A is * Hermitian* (or * self-adjoint*)
if .

Note that a Hermitian matrix is automatically square.

For real matrices, , and the definition above is just the definition of a symmetric matrix.

* Example.* Here are examples of Hermitian
matrices:

It is no accident that the diagonal entries are real numbers --- see the result that follows.

Here's a table of the correspondences between the real and complex cases:

* Proposition.* Let A be a Hermitian matrix.

(a) The diagonal elements of A are real numbers, and elements on opposite sides of the main diagonal are conjugates.

(b) The eigenvalues of a Hermitian matrix are real numbers.

(c) Eigenvectors of A corresponding to different eigenvalues are orthogonal.

* Proof.* (a) Since , I have . This shows
that elements on opposite sides of the main diagonal are conjugates.

Taking , I have

But a complex number is equal to its conjugate if and only if it's a real number, so is real.

(b) Suppose A is Hermitian and is an eigenvalue of A with eigenvector v. Then

Therefore, --- but a number that equals its complex conjugate must be real.

(c) Suppose is an eigenvalue of A with eigenvector u and is an eigenvalue of A with eigenvector v. Then

implies , so if the eigenvalues are different, then .

* Example.* Let

Show that the eigenvalues are real, and that eigenvectors for different eigenvalues are orthogonal.

The matrix is Hermitian. The characteristic polynomial is

The eigenvalues are real numbers: -4 and 2.

For -4, the eigenvector matrix is

is an eigenvector.

For 2, the eigenvector matrix is

is an eigenvector.

Note that

Thus, the eigenvectors are orthogonal.

Since real symmetric matrices are Hermitian, the previous results apply to them as well. I'll restate the previous result for the case of a symmetric matrix.

* Corollary.* Let A be a symmetric matrix.

(a) The elements on opposite sides of the main diagonal are equal.

(b) The eigenvalues of a symmetric matrix are real numbers.

(c) Eigenvectors of A corresponding to different eigenvalues are orthogonal.

* Example.* Consider the symmetric matrix

The characteristic polynomial is .

Note that the eigenvalues are real numbers.

For , an eigenvector is .

For , an eigenvector is .

Since , the eigenvectors are orthogonal.

* Example.* A real symmetric matrix A has
eigenvalues 1 and 3.

is an eigenvector corresponding to the eigenvalue 1.

(a) Find an eigenvector corresponding to the eigenvalue 3.

Let be an eigenvector corresponding to the eigenvalue 3.

Since eigenvectors for different eigenvalues of a symmetric matrix must be orthogonal, I have

So, for example, is a solution.

(b) Find A.

From (a), a diagonalizing matrix and the corresponding diagonal matrix are

Now , so

Note that the result is indeed symmetric.

* Example.* Let , and consider the Hermitian matrix

Compute the characteristic polynomial of A, and show directly that the eigenvalues must be real numbers.

The discriminant is

Since this is a sum of squares, it can't be negative. Hence, the roots of the characteristic polynomial --- the eigenvalues --- must be real numbers.

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Copyright 2014 by Bruce Ikenaga