##### 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. Understand what diagonalizability and multiplicity have to say about similarity.
4. Recipes: diagonalize a matrix, quickly compute powers of a matrix by diagonalization.
5. Pictures: the geometry of diagonal matrices, why a shear is not diagonalizable.
6. Theorem: the diagonalization theorem (two variants).
7. 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. In Section 5.3, we saw that similar matrices behave in the same way, with respect to different coordinate systems. Therefore, if a matrix is similar to a diagonal matrix, it is also relatively easy to understand. This section is devoted to the question: “When is a matrix similar to a diagonal matrix?”

# Subsection5.4.1Diagonalizability

Before answering the above question, first we give it a 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 by this proposition in Section 5.3.

##### 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 in Section 5.3, if then so it is also easy to take powers of diagonalizable matrices. This will be very important in applications to difference equations in Section 5.6.

##### 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:

Therefore, we know from Section 5.3 that a diagonalizable matrix simply scales the “axes” with respect to a different coordinate system. Indeed, if are linearly independent eigenvectors of an matrix then scales the -direction by the eigenvalue

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:
##### Similarity and multiplicity

Recall from this fact in Section 5.3 that similar matrices have the same eigenvalues. It turns out that both notions of multiplicity of an eigenvalue are preserved under similarity.

##### Proof

For instance, the four matrices in this example are not similar to each other, because the algebraic and/or geometric multiplicities of the eigenvalues do not match up. Or, combined with the above theorem, we see that a diagonalizable matrix cannot be similar to a non-diagonalizable one, because the algebraic and geometric multiplicities of such matrices cannot both coincide.

The converse of the theorem is false: there exist matrices whose eigenvectors have the same algebraic and geometric multiplicities, but which are not similar. See the example below. However, for and matrices whose characteristic polynomial has no complex (non-real) roots, the converse of the theorem is true. (We will handle the case of complex roots in Section 5.5.)

On the other hand, suppose that and are diagonalizable matrices with the same characteristic polynomial. Since the geometric multiplicities of the eigenvalues coincide with the algebraic multiplicities, which are the same for and we conclude that there exist linearly independent eigenvectors of each matrix, all of which have the same eigenvalues. This shows that and are both similar to the same diagonal matrix. Using the transitivity property of similar matrices, this shows:

Diagonalizable matrices are similar if and only if they have the same characteristic polynomial, or equivalently, the same eigenvalues with the same algebraic multiplicities.