MathTest

## Eigenvalues and eigenvectors of matrices

Template:See also

Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices. Furthermore, linear transformations can be represented using matrices, which is especially common in numerical and computational applications.

 File:Eigenvalue equation.svg
Consider Template:Mvar-dimensional vectors that are formed as a list of Template:Mvar scalars, such as the three-dimensional vectors

# x

[

1

3

4

]

and

# y

[

−
20

−
60

−
80

]

.

{\displaystyle x={\begin{bmatrix}1\\3\\4\end{bmatrix}}\quad {\mbox{and}}\quad y={\begin{bmatrix}-20\\-60\\-80\end{bmatrix}}.}

These vectors are said to be scalar multiples of each other, or parallel or collinear, if there is a scalar Template:Mvar such that

# x
λ
y
.

{\displaystyle x=\lambda y.}

In this case Template:Mvar = −1/20.

Now consider the linear transformation of Template:Mvar-dimensional vectors defined by an Template:Mvar by Template:Mvar matrix Template:Mvar,

A
# v
w
,

{\displaystyle Av=w,}

or

[

A

11

A

12

…

A

1
n

A

21

A

22

…

A

2
n

⋮

⋮

⋱

⋮

A

n
1

A

n
2

…

A

n
n

]

[

v

1

v

2

⋮

v

n

]

=

[

w

1

w

2

⋮

w

n

]

{\displaystyle {\begin{bmatrix}A\_{11}&A\_{12}&\ldots &A\_{1n}\\A\_{21}&A\_{22}&\ldots &A\_{2n}\\\vdots &\vdots &\ddots &\vdots \\A\_{n1}&A\_{n2}&\ldots &A\_{nn}\\\end{bmatrix}}{\begin{bmatrix}v\_{1}\\v\_{2}\\\vdots \\v\_{n}\end{bmatrix}}={\begin{bmatrix}w\_{1}\\w\_{2}\\\vdots \\w\_{n}\end{bmatrix}}}

where, for each row,

w

i

=

A

i
1

v

1

A

i
2

v

2

⋯

A

i
n

v

n

=

∑

# j
1

n

A

i
j

v

j

{\displaystyle w\_{i}=A\_{i1}v\_{1}+A\_{i2}v\_{2}+\cdots +A\_{in}v\_{n}=\sum \_{j=1}^{n}A\_{ij}v\_{j}}

.
If it occurs that Template:Mvar and Template:Mvar are scalar multiples, that is if
Template:NumBlk
then Template:Mvar is an **eigenvector** of the linear transformation Template:Mvar and the scale factor Template:Mvar is the **eigenvalue** corresponding to that eigenvector. Equation (Template:EquationNote) is the **eigenvalue equation** for the matrix Template:Mvar.

Equation (Template:EquationNote) can be stated equivalently as
Template:NumBlk
where Template:Mvar is the Template:Mvar by Template:Mvar identity matrix.

### Eigenvalues and the characteristic polynomial

*Main article:* Characteristic polynomial

Equation (Template:EquationNote) has a non-zero solution *v* if and only if the determinant of the matrix (*A* − *λI*) is zero. Therefore, the eigenvalues of *A* are values of *λ* that satisfy the equation
Template:NumBlk
Using Leibniz' rule for the determinant, the left hand side of Equation (Template:EquationNote) is a polynomial function of the variable *λ* and the degree of this polynomial is *n*, the order of the matrix *A*. Its coefficients depend on the entries of *A*, except that its term of degree *n* is always (−1)*n**λ**n*. This polynomial is called the *characteristic polynomial* of *A*. Equation (Template:EquationNote) is called the *characteristic equation* or the *secular equation* of *A*.

The fundamental theorem of algebra implies that the characteristic polynomial of an *n* by *n* matrix *A*, being a polynomial of degree *n*, can be factored into the product of *n* linear terms,
Template:NumBlk
where each *λ**i* may be real but in general is a complex number. The numbers *λ*1, *λ*2, ... *λ**n*, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of *A*.

As a brief example, which is described in more detail in the examples section later, consider the matrix

# M

[

2

1

1

2

]

.

{\displaystyle M={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}

Taking the determinant of (*M* − *λI*), the characteristic polynomial of *M* is

M
−
λ
I

=

2
−
λ

1

1

2
−
λ

=
3
−
4
λ

λ

2

.

{\displaystyle |M-\lambda I|={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.}

Setting the characteristic polynomial equal to zero, it has roots at Template:Nowrap and Template:Nowrap, which are the two eigenvalues of *M*. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of *v* in the equation *Mv* = *λv*. In this example, the eigenvectors are any non-zero scalar multiples of

v

# λ
1

=

[

1

−
1

]

,

v

# λ
3

=

[

1

1

]

.

{\displaystyle v\_{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad v\_{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}.}

If the entries of the matrix *A* are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have non-zero imaginary parts. The entries of the corresponding eigenvectors therefore may also have non-zero imaginary parts. Similarly, the eigenvalues may be irrational numbers even if all the entries of *A* are rational numbers or even if they are all integers. However, if the entries of *A* are all algebraic numbers, which include the rationals, the eigenvalues are complex algebraic numbers.

The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the intermediate value theorem at least one of the roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.

### Algebraic multiplicity

Let *λ**i* be an eigenvalue of an *n* by *n* matrix *A*. The **algebraic multiplicity** *μ**A*(*λ**i*) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer *k* such that (*λ* − *λ**i*)*k* divides evenly that polynomial.

Suppose a matrix *A* has dimension *n* and *d* ≤ *n* distinct eigenvalues. Whereas Equation (Template:EquationNote) factors the characteristic polynomial of *A* into the product of *n* linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of *d* terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,

A
−
λ
I

=
(

λ

1

−
λ

)

μ

A

(

λ

1

)

(

λ

2

−
λ

)

μ

A

(

λ

2

)

⋯
(

λ

d

−
λ

)

μ

A

(

λ

d

)

.

{\displaystyle |A-\lambda I|=(\lambda \_{1}-\lambda )^{\mu \_{A}(\lambda \_{1})}(\lambda \_{2}-\lambda )^{\mu \_{A}(\lambda \_{2})}\cdots (\lambda \_{d}-\lambda )^{\mu \_{A}(\lambda \_{d})}.}

If *d* = *n* then the right hand side is the product of *n* linear terms and this is the same as Equation (Template:EquationNote). The size of each eigenvalue's algebraic multiplicity is related to the dimension *n* as

1

≤

μ

A

(

λ

i

)
≤
n
,

μ

A

=

∑

# i
1

d

μ

A

(

λ

i

)

=
n
.

{\displaystyle {\begin{aligned}1&\leq \mu \_{A}(\lambda \_{i})\leq n,\\\mu \_{A}&=\sum \_{i=1}^{d}{\mu \_{A}(\lambda \_{i})}=n.\end{aligned}}}

If *μ**A*(*λ**i*) = 1, then *λ**i* is said to be a *simple eigenvalue*. If *μ**A*(*λ**i*) equals the geometric multiplicity of *λ**i*, *γ**A*(*λ**i*), defined in the next section, then *λ**i* is said to be a *semisimple eigenvalue*.

### Eigenspaces, geometric multiplicity, and the eigenbasis for matrices

Given a particular eigenvalue *λ* of the *n* by *n* matrix *A*, define the set *E* to be all vectors ***v*** that satisfy Equation (Template:EquationNote),

# E
{

v

:
(
A
−
λ
I
)

v

=
0
}
.

{\displaystyle E=\{\mathbf {v} :(A-\lambda I)\mathbf {v} =0\}.}

On one hand, this set is precisely the kernel or nullspace of the matrix (*A* − *λI*). On the other hand, by definition, any non-zero vector that satisfies this condition is an eigenvector of *A* associated with *λ*. So, the set *E* is the union of the zero vector with the set of all eigenvectors of *A* associated with *λ*, and *E* equals the nullspace of (*A* − *λI*). *E* is called the **eigenspace** or **characteristic space** of *A* associated with *λ*. In general *λ* is a complex number and the eigenvectors are complex *n* by 1 matrices. A property of the nullspace is that it is a linear subspace, so *E* is a linear subspace of ℂ*n*.

Because the eigenspace *E* is a linear subspace, it is closed under addition. That is, if two vectors *u* and *v* belong to the set *E*, written Template:Nowrap, then Template:Nowrap or equivalently *A*(*u* + *v*) = *λ*(*u* + *v*). This can be checked using the distributive property of matrix multiplication. Similarly, because *E* is a linear subspace, it is closed under scalar multiplication. That is, if Template:Nowrap and α is a complex number, Template:Nowrap or equivalently *A*(α*v*) = *λ*(*αv*). This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as *u* + *v* and *αv* are not zero, they are also eigenvectors of *A* associated with *λ*.

The dimension of the eigenspace *E* associated with *λ*, or equivalently the maximum number of linearly independent eigenvectors associated with *λ*, is referred to as the eigenvalue's **geometric multiplicity** *γ**A*(*λ*). Because *E* is also the nullspace of (*A* − *λI*), the geometric multiplicity of *λ* is the dimension of the nullspace of (*A* − *λI*), also called the *nullity* of (*A* − *λI*), which relates to the dimension and rank of (*A* − *λI*) as

γ

A

(
λ
# )
n
−
rank
⁡
(
A
−
λ
I
)
.

{\displaystyle \gamma \_{A}(\lambda )=n-\operatorname {rank} (A-\lambda I).}

Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed *n*.

1
≤

γ

A

(
λ
)
≤

μ

A

(
λ
)
≤
n

{\displaystyle 1\leq \gamma \_{A}(\lambda )\leq \mu \_{A}(\lambda )\leq n}

The condition that *γ**A*(*λ*) ≤ *μ**A*(*λ*) can be proven by considering a particular eigenvalue *ξ* of *A* and diagonalizing the first *γ**A*(*ξ*) columns of *A* with respect to the eigenvectors of *ξ*, described in a later section. The resulting similar matrix *B* is block upper triangular, with its top left block being the diagonal matrix *ξI**γ**A*(*ξ*). As a result, the characteristic polynomial of *B* will have a factor of (*ξ* − *λ*)*γ**A*(*ξ*). The other factors of the characteristic polynomial of *B* are not known, so the algebraic multiplicity of *ξ* as an eigenvalue of *B* is no less than the geometric multiplicity of *ξ* as an eigenvalue of *A*. The last element of the proof is the property that similar matrices have the same characteristic polynomial.

Suppose *A* has *d* ≤ *n* distinct eigenvalues *λ*1, *λ*2, ..., *λ**d*, where the geometric multiplicity of *λ**i* is *γ**A*(*λ**i*). The total geometric multiplicity of *A*,

γ

A

=

∑

# i
1

d

γ

A

(

λ

i

)
,

d

≤

γ

A

≤
n
,

{\displaystyle {\begin{aligned}\gamma \_{A}&=\sum \_{i=1}^{d}\gamma \_{A}(\lambda \_{i}),\\d&\leq \gamma \_{A}\leq n,\end{aligned}}}

is the dimension of the union of all the eigenspaces of *A'*s eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of *A*. If *γ**A* = *n*, then

* The union of the eigenspaces of all of *A'*s eigenvalues is the entire vector space ℂ*n*
* A basis of ℂ*n* can be formed from *n* linearly independent eigenvectors of *A*; such a basis is called an **eigenbasis**
* Any vector in ℂ*n* can be written as a linear combination of eigenvectors of *A*

### Additional properties of eigenvalues

Let *A* be an arbitrary *n* by *n* matrix of complex numbers with eigenvalues *λ*1, *λ*2, ..., *λ**n*. Each eigenvalue appears *μ**A*(*λ**i*) times in this list, where *μ**A*(*λ**i*) is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:

* The trace of *A*, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,

tr
⁡
(
A

# i
1

n

A

i
i

=

∑

# i
1

n

λ

i

=

λ

1

λ

2

⋯

λ

n

.

{\displaystyle \operatorname {tr} (A)=\sum \_{i=1}^{n}A\_{ii}=\sum \_{i=1}^{n}\lambda \_{i}=\lambda \_{1}+\lambda \_{2}+\cdots +\lambda \_{n}.}

* The determinant of *A* is the product of all its eigenvalues,

det
(
A

# i
1

n

λ

i

=

λ

1

λ

2

⋯

λ

n

.

{\displaystyle \det(A)=\prod \_{i=1}^{n}\lambda \_{i}=\lambda \_{1}\lambda \_{2}\cdots \lambda \_{n}.}

* The eigenvalues of the *k*th power of *A*, i.e. the eigenvalues of *A**k*, for any positive integer *k*, are *λ*1*k*, *λ*2*k*, ..., *λ**n**k*.
* The matrix *A* is invertible if and only if every eigenvalue is nonzero.
* If *A* is invertible, then the eigenvalues of *A*−1 are 1/*λ*1, 1/*λ*2, ..., 1/*λ**n* and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original, the eigenvalues share the same algebraic multiplicity.
* If *A* is equal to its conjugate transpose *A*\*, or equivalently if *A* is Hermitian, then every eigenvalue is real. The same is true of any symmetric real matrix.
* If *A* is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
* If *A* is unitary, every eigenvalue has absolute value |*λ**i*| = 1.

### Left and right eigenvectors

Template:See also

Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a **right eigenvector**, namely a *column* vector that *right* multiples the *n* by *n* matrix *A* in the defining equation, Equation (Template:EquationNote),

A
# v
λ
v
.

{\displaystyle Av=\lambda v.}

The eigenvalue and eigenvector problem can also be defined for *row* vectors that *left* multiply matrix *A*. In this formulation, the defining equation is

u
# A
κ
u
,

{\displaystyle uA=\kappa u,}

where κ is a scalar and *u* is a 1 by *n* matrix. Any row vector *u* satisfying this equation is called a **left eigenvector** of *A* and κ is its associated eigenvalue. Taking the transpose of this equation,

A

T

u

T

=
κ

u

T

.

{\displaystyle A^{T}u^{T}=\kappa u^{T}.}

Comparing this equation to Equation (Template:EquationNote), it follows immediately that a left eigenvector of *A* is the same as the transpose of a right eigenvector of *A*T, with the same eigenvalue. Furthermore, since the characteristic polynomial of *A*T is the same as the characteristic polynomial of *A*, the eigenvalues of the left eigenvectors of *A* are the same as the eigenvalues of the right eigenvectors of *A*T.

### Diagonalization and the eigendecomposition

*Main article:* Eigendecomposition of a matrix

Suppose the eigenvectors of *A* form a basis, or equivalently *A* has *n* linearly independent eigenvectors *v*1, *v*2, ..., *v**n* with associated eigenvalues *λ*1, *λ*2, ..., *λ**n*. The eigenvalues need not be distinct. Define a square matrix *Q* whose columns are the *n* linearly independent eigenvectors of *A*,

# Q

[

v
1

v
2

⋯

v
n

]

.

{\displaystyle Q = \begin{bmatrix} v\_1 & v\_2 & \cdots & v\_n \end{bmatrix}.}

Since each column of *Q* is an eigenvector of *A*, right multiplying *A* by *Q* scales each column of *Q* by its associated eigenvalue,

A
# Q

[

λ
1

v
1

λ
2

v
2

⋯

λ
n

v
n

]

.

{\displaystyle AQ = \begin{bmatrix} \lambda\_1 v\_1 & \lambda\_2 v\_2 & \cdots & \lambda\_n v\_n \end{bmatrix}.}

With this in mind, define a diagonal matrix Λ where each diagonal element Λ*ii* is the eigenvalue associated with the *i*th column of *Q*. Then

A
# Q
Q
Λ
.

{\displaystyle AQ = Q\Lambda.}

Because the columns of *Q* are linearly independent, Q is invertible. Right multiplying both sides of the equation by *Q*−1,

# A
Q
Λ

Q

−
1

,

{\displaystyle A = Q\Lambda Q^{-1},}

or by instead left multiplying both sides by *Q*−1,

Q

−
1

A
# Q
Λ
.

{\displaystyle Q^{-1}AQ = \Lambda.}

*A* can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix *A* is said to be *similar* to the diagonal matrix Λ or *diagonalizable*. The matrix *Q* is the change of basis matrix of the similarity transformation. Essentially, the matrices *A* and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ.

Conversely, suppose a matrix *A* is diagonalizable. Let *P* be a non-singular square matrix such that *P*−1*AP* is some diagonal matrix *D*. Left multiplying both by *P*, *AP* = *PD*. Each column of *P* must therefore be an eigenvector of *A* whose eigenvalue is the corresponding diagonal element of *D*. Since the columns of *P* must be linearly independent for *P* to be invertible, there exist *n* linearly independent eigenvectors of *A*. It then follows that the eigenvectors of *A* form a basis if and only if *A* is diagonalizable.

A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix *A* has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.

### Variational characterization

*Main article:* Min-max theorem

In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of

H

{\displaystyle H}

 is the maximum value of the quadratic form

x

T

H
x

/

x

T

x

{\displaystyle x^{\mathsf {T}}Hx/x^{\mathsf {T}}x}

. A value of

x

{\displaystyle x}

 that realizes that maximum, is an eigenvector.

### Matrix examples

#### Two-dimensional matrix example

 File:Eigenvectors.gif
Consider the matrix

# A

[

2

1

1

2

]

.

{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}

The figure on the right shows the effect of this transformation on point coordinates in the plane.
The eigenvectors *v* of this transformation satisfy Equation (Template:EquationNote), and the values of *λ* for which the determinant of the matrix (*A* − *λI*) equals zero are the eigenvalues.

Taking the determinant to find characteristic polynomial of *A*,

A
−
λ
I

=

[

2

1

1

2

]

−
λ

[

1

0

0

1

]

=

2
−
λ

1

1

2
−
λ

,

=
3
−
4
λ

λ

2

.

Setting the characteristic polynomial equal to zero, it has roots at Template:Nowrap and Template:Nowrap, which are the two eigenvalues of *A*.

For Template:Nowrap, Equation (Template:EquationNote) becomes,

(
A
−
I
)

v

# λ
1

=

[

1

1

1

1

]

[

v

1

v

2

]

=

[

0

0

]

.

{\displaystyle (A-I)v\_{\lambda =1}={\begin{bmatrix}1&1\\1&1\end{bmatrix}}{\begin{bmatrix}v\_{1}\\v\_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.}

Any non-zero vector with *v*1 = −*v*2 solves this equation. Therefore,

v

# λ
1

=

[

1

−
1

]

{\displaystyle v\_{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}}}

is an eigenvector of *A* corresponding to *λ* = 1, as is any scalar multiple of this vector.

For Template:Nowrap, Equation (Template:EquationNote) becomes

(
A
−
3
I
)

v

# λ
3

=

[

−
1

1

1

−
1

]

[

v

1

v

2

]

=

[

0

0

]

.

{\displaystyle (A-3I)v\_{\lambda =3}={\begin{bmatrix}-1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}v\_{1}\\v\_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}.}

Any non-zero vector with *v*1 = *v*2 solves this equation. Therefore,

v

# λ
3

=

[

1

1

]

{\displaystyle v\_{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}}

is an eigenvector of *A* corresponding to *λ* = 3, as is any scalar multiple of this vector.

Thus, the vectors *v**λ*=1 and *v**λ*=3 are eigenvectors of *A* associated with the eigenvalues Template:Nowrap and Template:Nowrap, respectively.

# A

[

2

0

0

0

3

4

0

4

9

]

.

{\displaystyle A={\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}.}

The characteristic polynomial of *A* is

A
−
λ
I

=

[

2

0

0

0

3

4

0

4

9

]

−
λ

[

1

0

0

0

1

0

0

0

1

]

=

2
−
λ

0

0

0

3
−
λ

4

0

4

9
−
λ

,

=
(
2
−
λ
)

[

(
3
−
λ
)
(
9
−
λ
)
−
16

]

=
−

λ

3

14

λ

2

−
35
λ

The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of *A*. These eigenvalues correspond to the eigenvectors

[
1

0

0

]

T

,

{\displaystyle [1\;0\;0]^{\mathsf {T}},}

[
0

2

−
1

]

T

,

{\displaystyle [0\;2\;-1]^{\mathsf {T}},}

 and

[
0

1

2

]

T

{\displaystyle [0\;1\;2]^{\mathsf {T}}}

, or any non-zero multiple thereof.

# A

[

0

1

0

0

0

1

1

0

0

]

.

{\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.}

This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − *λ*3, whose roots are

λ
1

=
1

{\displaystyle \lambda\_1 = 1 }

λ

2

=
−
1

/

2

i

3

/

2

{\displaystyle \lambda \_{2}=-1/2+\mathbf {i} {\sqrt {3}}/2}

λ

3

=

λ

2

∗

=
−
1

/

2
−

i

3

/

2

{\displaystyle \lambda \_{3}=\lambda \_{2}^{\*}=-1/2-\mathbf {i} {\sqrt {3}}/2}

where **i** =

−
1

{\displaystyle \sqrt{-1}}

 is the imaginary unit.

For the real eigenvalue *λ*1 = 1, any vector with three equal non-zero entries is an eigenvector. For example,

A

[

5

5

5

]

=

[

5

5

5

]

=
1
⋅

[

5

5

5

]

.

{\displaystyle A{\begin{bmatrix}5\\5\\5\end{bmatrix}}={\begin{bmatrix}5\\5\\5\end{bmatrix}}=1\cdot {\begin{bmatrix}5\\5\\5\end{bmatrix}}.}

For the complex conjugate pair of imaginary eigenvalues, note that

λ

2

λ

3

=
1
,

λ

2

2

=

λ

3

,

λ

3

2

=

λ

2

.

{\displaystyle \lambda \_{2}\lambda \_{3}=1,\quad \lambda \_{2}^{2}=\lambda \_{3},\quad \lambda \_{3}^{2}=\lambda \_{2}.}

Then

A

[

1

λ

2

λ

3

]

=

[

λ

2

λ

3

1

]

=

λ

2

⋅

[

1

λ

2

λ

3

]

,

{\displaystyle A{\begin{bmatrix}1\\\lambda \_{2}\\\lambda \_{3}\end{bmatrix}}={\begin{bmatrix}\lambda \_{2}\\\lambda \_{3}\\1\end{bmatrix}}=\lambda \_{2}\cdot {\begin{bmatrix}1\\\lambda \_{2}\\\lambda \_{3}\end{bmatrix}},}

and

A

[

1

λ

3

λ

2

]

=

[

λ

3

λ

2

1

]

=

λ

3

⋅

[

1

λ

3

λ

2

]

.

{\displaystyle A{\begin{bmatrix}1\\\lambda \_{3}\\\lambda \_{2}\end{bmatrix}}={\begin{bmatrix}\lambda \_{3}\\\lambda \_{2}\\1\end{bmatrix}}=\lambda \_{3}\cdot {\begin{bmatrix}1\\\lambda \_{3}\\\lambda \_{2}\end{bmatrix}}.}

Therefore, the other two eigenvectors of *A* are complex and are

v

λ

2

=
[
1

λ

2

λ

3

]

T

{\displaystyle v\_{\lambda \_{2}}=[1\;\lambda \_{2}\;\lambda \_{3}]^{\mathrm {T} }}

 and

v

λ

3

=
[
1

λ

3

λ

2

]

T

{\displaystyle v\_{\lambda \_{3}}=[1\;\lambda \_{3}\;\lambda \_{2}]^{\mathrm {T} }}

 with eigenvalues *λ*2 and *λ*3, respectively. Note that the two complex eigenvectors also appear in a complex conjugate pair,

v

λ

2

=

v

λ

3

∗

.

{\displaystyle v\_{\lambda \_{2}}=v\_{\lambda \_{3}}^{\*}.}

#### Diagonal matrix example

Matrices with entries only along the main diagonal are called *diagonal matrices*. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix

# A

[

1

0

0

0

2

0

0

0

3

]

.

{\displaystyle A={\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}}.}

The characteristic polynomial of *A* is

A
−
λ
I

=
(
1
−
λ
)
(
2
−
λ
)
(
3
−
λ
)
,

{\displaystyle |A-\lambda I|=(1-\lambda )(2-\lambda )(3-\lambda ),}

which has the roots Template:Nowrap, Template:Nowrap, and Template:Nowrap. These roots are the diagonal elements as well as the eigenvalues of *A*.

Each diagonal element corresponds to an eigenvector whose only non-zero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,

v

λ

1

=

[

1

0

0

]

,

v

λ

2

=

[

0

1

0

]

,

v

λ

3

=

[

0

0

1

]

,

{\displaystyle v\_{\lambda \_{1}}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad v\_{\lambda \_{2}}={\begin{bmatrix}0\\1\\0\end{bmatrix}},\quad v\_{\lambda \_{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}

respectively, as well as scalar multiples of these vectors.

#### Triangular matrix example

A matrix whose elements above the main diagonal are all zero is called a *lower triangular matrix*, while a matrix whose elements below the main diagonal are all zero is called an *upper triangular matrix*. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.

Consider the lower triangular matrix,

# A

[

1

0

0

1

2

0

2

3

3

]

.

{\displaystyle A={\begin{bmatrix}1&0&0\\1&2&0\\2&3&3\end{bmatrix}}.}

The characteristic polynomial of *A* is

A
−
λ
I

=
(
1
−
λ
)
(
2
−
λ
)
(
3
−
λ
)
,

{\displaystyle |A-\lambda I|=(1-\lambda )(2-\lambda )(3-\lambda ),}

which has the roots Template:Nowrap, Template:Nowrap, and Template:Nowrap. These roots are the diagonal elements as well as the eigenvalues of *A*.

These eigenvalues correspond to the eigenvectors,

v

λ

1

=

[

1

−
1

1

/

2

]

,

v

λ

2

=

[

0

1

−
3

]

,

v

λ

3

=

[

0

0

1

]

,

{\displaystyle v\_{\lambda \_{1}}={\begin{bmatrix}1\\-1\\1/2\end{bmatrix}},\quad v\_{\lambda \_{2}}={\begin{bmatrix}0\\1\\-3\end{bmatrix}},\quad v\_{\lambda \_{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}

respectively, as well as scalar multiples of these vectors.

#### Matrix with repeated eigenvalues example

As in the previous example, the lower triangular matrix

# A

[

2

0

0

0

1

2

0

0

0

1

3

0

0

0

1

3

]

,

{\displaystyle A={\begin{bmatrix}2&0&0&0\\1&2&0&0\\0&1&3&0\\0&0&1&3\end{bmatrix}},}

has a characteristic polynomial that is the product of its diagonal elements,

A
−
λ
I

=

2
−
λ

0

0

0

1

2
−
λ

0

0

0

1

3
−
λ

0

0

0

1

3
−
λ

=
(
2
−
λ

)

2

(
3
−
λ

)

2

.

{\displaystyle |A-\lambda I|\;=\;{\begin{vmatrix}2-\lambda &0&0&0\\1&2-\lambda &0&0\\0&1&3-\lambda &0\\0&0&1&3-\lambda \end{vmatrix}}=(2-\lambda )^{2}(3-\lambda )^{2}.}

The roots of this polynomial, and hence the eigenvalues, are 2 and 3.
The *algebraic multiplicity* of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of each distinct eigenvalue is *μ**A* = 4 = *n*, the order of the characteristic polynomial and the dimension of *A*.

On the other hand, the *geometric multiplicity* of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector [0 1 −1 1]T and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector [0 0 0 1]T. The total geometric multiplicity *γ**A* is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.

1. ↑ Cornell University Department of Mathematics (2016) *Lower-Level Courses for Freshmen and Sophomores*. Accessed on 2016-03-27.
2. ↑ University of Michigan Mathematics (2016) *Math Course Catalogue* Template:Webarchive. Accessed on 2016-03-27.
3. ↑ Cite error: Invalid `<ref>` tag; no text was provided for refs named `Herstein 1964 228–299`
4. ↑ Cite error: Invalid `<ref>` tag; no text was provided for refs named `Nering 1970 38`
5. ↑ Template:Harvtxt
6. ↑ 6.0 6.1 Cite error: Invalid `<ref>` tag; no text was provided for refs named `Nering 1970 107`
7. ↑ Template:Harvtxt
8. ↑ 8.0 8.1 Cite error: Invalid `<ref>` tag; no text was provided for refs named `Golub 1996 316`
9. ↑ Cite error: Invalid `<ref>` tag; no text was provided for refs named `Anton 1987 305,307`
10. ↑ 10.0 10.1 Template:Harvtxt
11. ↑ Template:Harvtxt
12. ↑ Template:Harvtxt
13. ↑ Template:Harvtxt
14. ↑ Template:Harvtxt