##### Objectives
1. Learn two main criteria for a matrix to be diagonalizable.
2. Develop a library of examples of matrices that are and are not diagonalizable.
3. Recipes: diagonalize a matrix, quickly compute powers of a matrix by diagonalization.
4. Pictures: the geometry of diagonal matrices, why a shear is not diagonalizable.
5. Theorem: the diagonalization theorem (two variants).
6. Vocabulary words: diagonalizable, algebraic multiplicity, geometric multiplicity.

Diagonal matrices are the easiest kind of matrices to understand: they just scale the coordinate directions by their diagonal entries. This section is devoted to the question: “When is a matrix similar to a diagonal matrix?” We will see that the algebra and geometry of such a matrix is relatively easy to understand.

# Subsection5.4.1Diagonalizability

First we make precise what we mean when we say two matrices are “similar”.

##### Definition

Two matrices and are similar if there exists an invertible matrix such that

If two matrices are similar, then their powers are similar as well.

##### Proof

First note that

Next we have

The pattern is clear.

In this chapter, we will determine when a matrix is similar to a diagonal matrix. This property is important enough to deserve its own name.

##### Definition

An matrix is diagonalizable if it is similar to a diagonal matrix: that is, if there exists an invertible matrix and a diagonal matrix such that

##### Example

Any diagonal matrix is is diagonalizable because it is similar to itself. For instance,

##### Example

If a matrix is diagonalizable, and if is similar to then is diagonalizable as well. Indeed, if for diagonal, and then

so is similar to

##### Powers of diagonalizable matrices

Multiplying diagonal matrices together just multiplies their diagonal entries:

Therefore, it is easy to take powers of a diagonal matrix:

By this fact, if then so it is also easy to take powers of diagonalizable matrices. This is often very important in applications.

##### Recipe: Compute powers of a diagonalizable matrix

If where is a diagonal matrix, then

A fundamental question about a matrix is whether or not it is diagonalizable. The following is the primary criterion for diagonalizability. It shows that diagonalizability is an eigenvalue problem.

##### Proof

By this fact in Section 5.1, if an matrix has distinct eigenvalues then a choice of corresponding eigenvectors is automatically linearly independent.

An matrix with distinct eigenvalues is diagonalizable.

##### Non-Uniqueness of Diagonalization

We saw in the above example that changing the order of the eigenvalues and eigenvectors produces a different diagonalization of the same matrix. There are generally many different ways to diagonalize a matrix, corresponding to different orderings of the eigenvalues of that matrix. The important thing is that the eigenvalues and eigenvectors have to be listed in the same order.

There are other ways of finding different diagonalizations of the same matrix. For instance, you can scale one of the eigenvectors by a constant

you can find a different basis entirely for an eigenspace of dimension at least etc.

In the above example, the (non-invertible) matrix is similar to the diagonal matrix Since is not invertible, zero is an eigenvalue by the invertible matrix theorem, so one of the diagonal entries of is necessarily zero. Also see this example below.

Here is the procedure we used in the above examples.

##### Recipe: Diagonalization

Let be an matrix. To diagonalize

1. Find the eigenvalues of using the characteristic polynomial.
2. For each eigenvalue of compute a basis for the -eigenspace.
3. If there are fewer than total vectors in all of the eigenspace bases then the matrix is not diagonalizable.
4. Otherwise, the vectors in the eigenspace bases are linearly independent, and for
where is the eigenvalue for

We will justify the linear independence assertion in part 4 in the proof of this theorem below.

The following point is often a source of confusion.

##### Diagonalizability has nothing to do with invertibility

Of the following matrices, the first is diagonalizable and invertible, the second is diagonalizable but not invertible, the third is invertible but not diagonalizable, and the fourth is neither invertible nor diagonalizable, as the reader can verify:

# Subsection5.4.2The Geometry of Diagonalizable Matrices¶ permalink

A diagonal matrix is easy to understand geometrically, as it just scales the coordinate axes:

A daigonalizable matrix is not much harder to understand geometrically. Indeed, if are linearly independent eigenvectors of an matrix then scales the -direction by the eigenvalue in other words, Since the vectors form a basis of this determines the action of on any vector in

##### Example

Consider the matrices

One can verify that see this example. Let and the columns of These are eigenvectors of with corresponding eigenvalues and

The matrix is diagonal: it scales the -direction by a factor of and the -direction by a factor of

If we write a vector in terms of the basis say, then it is easy to compute

Here we have used the fact that are eigenvectors of Since the resulting vector is still expressed in terms of the basis we can visualize what does to the vector it scales the -coordinate” by and the -coordinate” by

For instance, let We see from the grid on the right in the picture below that so

The picture illustrates the action of on the plane in the usual basis, and the action of on the plane in the -basis.

Now let We see from the grid on the right in the picture below that so

This is illustrated in the picture below.

In the following examples, we visualize the action of a diagonalizable matrix in terms of its dynamics. In other words, we start with a collection of vectors (drawn as points), and we see where they move when we multiply them by repeatedly.

# Subsection5.4.3Algebraic and Geometric Multiplicity

In this subsection, we give a variant of the diagonalization theorem that provides another criterion for diagonalizability. It is stated in the language of multiplicities of eigenvalues.

In algebra, we define the multiplicity of a root of a polynomial to be the number of factors of that divide For instance, in the polynomial

the root has multiplicity and the root has multiplicity

##### Definition

Let be an matrix, and let be an eigenvalue of

1. The algebraic multiplicity of is its multiplicity as a root of the characteristic polynomial of
2. The geometric multiplicity of is the dimension of the -eigenspace.

Since the -eigenspace of is its dimension is the number of free variables in the system of equations i.e., the number of columns without pivots in the matrix

We saw in the above examples that the algebraic and geometric multiplicities need not coincide. However, they do satisfy the following fundamental inequality, the proof of which is beyond the scope of this text.

In particular, if the algebraic multiplicity of is equal to then so is the geometric multiplicity.

If has an eigenvalue with algebraic multiplicity then the -eigenspace is a line.

We can use the theorem to give another criterion for diagonalizability (in addition to the diagonalization theorem).

##### Proof

The first part of the third statement simply says that the characteristic polynomial of factors completely into linear polynomials over the real numbers: in other words, there are no complex (non-real) roots. The second part of the third statement says in particular that for any diagonalizable matrix, the algebraic and geometric multiplicities coincide.

Let be a square matrix and let be an eigenvalue of If the algebraic multiplicity of does not equal the geometric multiplicity, then is not diagonalizable.

The examples at the beginning of this subsection illustrate the theorem. Here we give some general consequences for diagonalizability of and matrices.

##### Diagonalizability of 2 × 2 Matrices

Let be a matrix. There are four cases:

1. has two different eigenvalues. In this case, each eigenvalue has algebraic and geometric multiplicity equal to one. This implies is diagonalizable. For example:
2. has one eigenvalue of algebraic and geometric multiplicity To say that the geometric multiplicity is means that i.e., that every vector in is in the null space of This implies that is the zero matrix, so that is the diagonal matrix In particular, is diagonalizable. For example:
3. has one eigenvalue of algebraic multiplicity and geometric multiplicity In this case, is not diagonalizable, by part 3 of the theorem. For example, a shear:
4. has no eigenvalues. This happens when the characteristic polynomial has no real roots. In particular, is not diagonalizable. For example, a rotation: