Eigenvalues and eigenvectors

In this section, we will define the eigenvalues and eigenvectors of a matrix, and see how to compute them. More theoretical properties will be taken up in the next section.

1. A=\begin{bmatrix} -5 & 8\\-4 & 7 \end{bmatrix}
2. For each eigenvalue of $A$, find the corresponding eigenspace.
3. For the polynomial $p(x)=3x^2-x+2$ and $A$ from above, compute $p(A)$.

We start with the principal definition for this chapter.

Before going any further, perhaps we should convince you that such things ever happen at all. Understand the next example, but do not concern yourself with where the pieces come from. We will have methods soon enough to be able to discover these eigenvectors ourselves.

Sage can compute eigenvalues and eigenvectors of matrices. We will see shortly that there are subtleties involved with using these routines, but here is a quick example to begin with. These two commands should be enough to get you started with most of the early examples in this section. See the end of the section for more comprehensive advice.

For a square matrix, the methods .eigenvalues() and .eigenvectors_right() will produce what you expect, though the format of the eigenvector output requires some explanation. Here is from the start of this chapter.

The three eigenvalues we know are included in the output of eigenvalues(), though for some reason the eigenvalue $\lambda=2$ shows up twice.

The output of the eigenvectors_right() method is a list of triples. Each triple begins with an eigenvalue. This is followed by a list of eigenvectors for that eigenvalue. Notice the first eigenvector is identical to the one we described in . The eigenvector for $\lambda=0$ is different, but is just a scalar multiple of the one from . For $\lambda=2$, we now get two eigenvectors, and neither looks like either of the ones from . (Hint: try writing the eigenvectors from the example as linear combinations of the two in the Sage output.) An explanation of the the third part of each triple (an integer) will have to wait, though it can be optionally surpressed if desired.

in Sage, so do not try to use this as a name for your eigenvalues.

hints at a number of intriguing properties, and there are many more. We will explore the general properties of eigenvalues and eigenvectors in , but in this section we will concern ourselves with the question of actually computing eigenvalues and eigenvectors. First we need a bit of background material on polynomials and matrices.

Condensed: the full section is in Beezer, A First Course in Linear Algebra.

A polynomial is a combination of powers, multiplication by scalar coefficients, and addition (with subtraction just being the inverse of addition). We never have occasion to divide when computing the value of a polynomial. So it is with matrices. We can add and subtract matrices, we can multiply matrices by scalars, and we can form powers of square matrices by repeated applications of matrix multiplication. We do not normally divide matrices (though sometimes we can multiply by an inverse). If a matrix is square, all the operations constituting a polynomial will preserve the size of the matrix. So it is natural to consider evaluating a polynomial with a matrix, effectively replacing the variable of the polynomial by a matrix. We will demonstrate with an example.

Condensed: the full section is in Beezer, A First Course in Linear Algebra.

Before we embark on computing eigenvalues and eigenvectors, we will prove that every matrix has at least one eigenvalue (and an eigenvector to go with it). Later, in , we will determine the maximum number of eigenvalues a matrix may have.

The determinant () will be a powerful tool in when it comes time to compute eigenvalues. However, it is possible, with some more advanced machinery, to compute eigenvalues without ever making use of the determinant. Sheldon Axler does just that in his book, Linear Algebra Done Right. Here and now, we give Axler's determinant-free proof that every matrix has an eigenvalue. The result is not too startling, but the proof is most enjoyable.

The proof of is constructive (it contains an unambiguous procedure that leads to an eigenvalue), but it is not meant to be practical. We will illustrate the theorem with an example, the purpose being to provide a companion for studying the proof and not to suggest this is the best procedure for computing an eigenvalue.

Condensed: the full section is in Beezer, A First Course in Linear Algebra.

Fortunately, we need not rely on the procedure of each time we need an eigenvalue. It is the determinant, and specifically , that provides the main tool for computing eigenvalues. Here is an informal sequence of equivalences that is the key to determining the eigenvalues and eigenvectors of a matrix, A\vect{x}=\lambda\vect{x}\iff A\vect{x}-\lambda I_n\vect{x}=\zerovector\iff \left(A-\lambda I_n\right)\vect{x}=\zerovector

So, for an eigenvalue $\lambda$ and associated eigenvector $\vect{x}\neq\zerovector$, the vector $\vect{x}$ will be a nonzero element of the null space of $A-\lambda I_n$, while the matrix $A-\lambda I_n$ will be singular and therefore have zero determinant. These ideas are made precise in and , but for now this brief discussion should suffice as motivation for the following definition and example.

The characteristic polynomial is our main computational tool for finding eigenvalues, and will sometimes be used to aid us in determining the properties of eigenvalues.

Let us now turn our attention to the computation of eigenvectors.

Condensed: the full section is in Beezer, A First Course in Linear Algebra.

There are no theorems in this section, just a selection of examples meant to illustrate the range of possibilities for the eigenvalues and eigenvectors of a matrix. These examples can all be done by hand, though the computation of the characteristic polynomial would be very time-consuming and error-prone. It can also be difficult to factor an arbitrary polynomial, though if we were to suggest that most of our eigenvalues are going to be integers, then it can be easier to hunt for roots. These examples are meant to look similar to a concatenation of , and . First, we will sneak in a pair of definitions so we can illustrate them throughout this sequence of examples.

Since an eigenvalue $\lambda$ is a root of the characteristic polynomial, there is always a factor of $(x-\lambda)$, and the algebraic multiplicity is just the power of this factor in a factorization of $\charpoly{A}{x}$. So in particular, $\algmult{A}{\lambda}\geq 1$. Compare the definition of algebraic multiplicity with the next definition.

Every eigenvalue must have at least one eigenvector, so the associated eigenspace cannot be trivial, and so $\geomult{A}{\lambda}\geq 1$.

Condensed: the full section is in Beezer, A First Course in Linear Algebra.

In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \(\mathbf v\) of a linear transformation \(T\) is scaled by a constant factor \(\lambda\) when the linear transformation is applied to it: ⁠\(T \mathbf v = \lambda \mathbf v\)⁠. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \(\lambda\) (possibly a negative or complex number).

Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, then the eigenvector's direction is reversed.

The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all areas where linear algebra is applied, from geology to quantum mechanics. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation (feedback). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of the linear transformation, and the associated eigenvector is the steady state of the system.

For an \(n{\times}n\) matrix ⁠\(A\)⁠ and a nonzero ⁠\(n\)⁠-vector ⁠\(\mathbf v\)⁠, if multiplying ⁠\(A\)⁠ by \(\mathbf{v}\) (denoted ⁠\(A \mathbf v\)⁠) simply scales \(\mathbf{v}\) by a factor ⁠\(\lambda\)⁠, where ⁠\(\lambda\)⁠ is a scalar, then \(\mathbf{v}\) is called an eigenvector of ⁠\(A\)⁠, and ⁠\(\lambda\)⁠ is the corresponding eigenvalue. This relationship can be expressed as: ⁠\(A \mathbf v = \lambda \mathbf v\)⁠.

Given an ⁠\(n\)⁠-dimensional vector space and a choice of basis, there is a direct correspondence between linear transformations from the vector space into itself and ⁠\(n \times n\)⁠ square matrices. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of linear transformations, or the language of matrices.

Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German eigen (cognate with the English word own) for 'proper', 'characteristic', 'own'. Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.

In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation \[T(\mathbf{v}) = \lambda \mathbf{v},\] referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero, or complex.

The example here, based on the Mona Lisa, provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either.

Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like ⁠\(\tfrac{d}{dx}\)⁠, in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as \[\frac{d}{dx} e^{\lambda x} = \lambda e^{\lambda x}.\] Alternatively, the linear transformation could take the form of an n × n matrix, in which case the eigenvectors are n × 1 matrices.

If the linear transformation is expressed in the form of an n × n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication \[A \mathbf v = \lambda \mathbf v,\] where the eigenvector v is an n × 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix; for example, by diagonalizing it. Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen- is applied liberally when naming them:

Condensed: the full section is in Wikipedia.

Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.

In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.

In the early 19th century, Augustin-Louis Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in characteristic equation.

Later, Joseph Fourier used the work of Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of variables in his 1822 treatise The Analytic Theory of Heat (Théorie analytique de la chaleur). Charles-François Sturm elaborated on Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.

Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle, and Alfred Clebsch found the corresponding result for skew-symmetric matrices. Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability.

In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm-Liouville theory. Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.

At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Hermann von Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.

Condensed: the full section is in Wikipedia.

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

Consider two ⁠\(n\)⁠-dimensional vectors that are formed as a list of ⁠\(n\)⁠ scalars, such as the three-dimensional vectors \[\mathbf{x} = \begin{bmatrix} 1 \\ -3 \\ 4 \end{bmatrix} \quad \mbox{and} \quad \mathbf{y} = \begin{bmatrix} -20 \\ 60 \\ -80 \end{bmatrix}.\] These vectors are said to be scalar multiples of each other, or parallel, or collinear, if there is a scalar ⁠\(\lambda\)⁠ such that \[\mathbf y = \lambda \mathbf x.\] In this example, ⁠\(\lambda = -20\)⁠.

Now consider the linear transformation of ⁠\(n\)⁠-dimensional vectors defined by an ⁠\(n \times n\)⁠ matrix ⁠\(A\)⁠: \[A \mathbf v = \mathbf w,\] or \[\begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} & \cdots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} & A_{n2} & \cdots & A_{nn} \\ \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix}\] where, for each row, \[w_i = A_{i1} v_1 + A_{i2} v_2 + \cdots + A_{in} v_n = \sum_{j=1}^n A_{ij} v_j.\]

If it occurs that ⁠\(\mathbf v\)⁠ and ⁠\(\mathbf w\)⁠ are scalar multiples, that is, if

then ⁠\(\mathbf v\)⁠ is an eigenvector of the linear transformation ⁠\(A\)⁠ and the scale factor ⁠\(\lambda\)⁠ is the eigenvalue corresponding to that eigenvector. Equation (1) is the eigenvalue equation for the matrix ⁠\(A\)⁠.

Equation (1) can be stated equivalently as

where ⁠\(I\)⁠ is the ⁠\(n \times n\)⁠ identity matrix and ⁠\(\mathbf 0\)⁠ is the zero vector.

Equation (2) has a nonzero 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

Using the Leibniz formula for determinants, the left-hand side of equation (3) 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)λ. This polynomial is called the characteristic polynomial of A. Equation (3) is called the characteristic equation or secular equation of A.

The characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, has at most n complex number roots, which can be found by factoring the characteristic polynomial, or numerically by root finding. The characteristic polynomial can be factored into the product of n linear terms:

where the complex numbers λ₁, λ₂, ..., λ_n, each of which is an eigenvalue, may repeat. (The number of times an eigenvalue appears in the characteristic polynomial is known as its algebraic multiplicity.)

As a brief example, which is described in more detail in the examples section later, consider the matrix \[A = \begin{bmatrix} 2 & 1\\ 1 & 2 \end{bmatrix}.\] Taking the determinant of (A − λI), the characteristic polynomial of A is \[\det(A - \lambda I) = \begin{vmatrix} 2 - \lambda & 1 \\ 1 & 2 - \lambda \end{vmatrix} = 3 - 4\lambda + \lambda^2.\] Setting the characteristic polynomial equal to zero, it has roots at λ = 1 and λ = 3, which are the two eigenvalues of A. The eigenvectors corresponding to each eigenvalue λ can be found by solving for the components of v in the equation (A − λI)v = 0. In this example, the eigenvectors are any nonzero scalar multiples of \[\mathbf{v}_{\lambda=1} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \quad \mathbf{v}_{\lambda=3} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}.\]

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 nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero 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, then the eigenvalues must also be 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.

The spectrum of a matrix is the list of its eigenvalues, repeated according to their multiplicities; in a shorter notation, the set of its eigenvalues with their multiplicities indicated.

An important quantity associated with the spectrum of a matrix is the maximum absolute value of all of its eigenvalues. This is known as the spectral radius of the considered matrix.

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_i − λ) evenly divides that polynomial.

Suppose a matrix A has order n and d ≤ n distinct eigenvalues. Whereas equation (4) factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity: \[\det(A - \lambda I) = (\lambda_1 - \lambda)^{\mu_A(\lambda_1)}(\lambda_2 - \lambda)^{\mu_A(\lambda_2)} \cdots (\lambda_d - \lambda)^{\mu_A(\lambda_d)}.\] If d = n, then the right-hand side is the product of n linear terms, and this is the same as equation (4). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as \[\begin{aligned} 1 &\leq \mu_A(\lambda_i) \leq n, \\ \mu_A &= \sum_{i=1}^d \mu_A\left(\lambda_i\right) = n. \end{aligned}\] 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 (denoted by γ_A(λ_i) and defined in the next section), then λ_i is said to be a semisimple eigenvalue.

Given a particular eigenvalue ⁠\(\lambda\)⁠ of the ⁠\(n \times n\)⁠ matrix ⁠\(A\)⁠, define the set ⁠\(E\)⁠ to be all vectors ⁠\(\mathbf v\)⁠ that satisfy equation (2): \[E = \left\{ \mathbf{v} : \left( A - \lambda I \right) \mathbf{v} = \mathbf{0} \right\}.\] On one hand, ⁠\(E\)⁠ is precisely the kernel or nullspace of the matrix ⁠\(A - \lambda I\)⁠. On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of ⁠\(A\)⁠ associated with ⁠\(\lambda\)⁠; so ⁠\(E\)⁠ is the union of the zero vector with the set of all eigenvectors of ⁠\(A\)⁠ associated with ⁠\(\lambda\)⁠. The space ⁠\(E\)⁠ is called the eigenspace or characteristic space of ⁠\(A\)⁠ associated with ⁠\(\lambda\)⁠. In general, ⁠\(\lambda\)⁠ is a complex number and the eigenvectors are complex ⁠\(n \times 1\)⁠ matrices (column vectors). Because every nullspace is a linear subspace of the domain, ⁠\(E\)⁠ 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 ⁠\(\mathbf u\)⁠ and ⁠\(\mathbf v\)⁠ belong to the set ⁠\(E\)⁠, written ⁠\(\mathbf{u}, \mathbf{v} \in E\)⁠, then ⁠\(\mathbf{u} + \mathbf{v} \in E\)⁠, or equivalently ⁠\(A(\mathbf{u} + \mathbf{v}) = \lambda(\mathbf{u} + \mathbf{v})\)⁠. 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 ⁠\(\mathbf{v} \in E\)⁠ and ⁠\(\alpha \in \C\)⁠, then ⁠\(\alpha \mathbf{v} \in E\)⁠, or equivalently ⁠\(A(\alpha \mathbf{v}) = \lambda(\alpha \mathbf{v})\)⁠. This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as ⁠\(\mathbf{u} + \mathbf{v}\)⁠ and ⁠\(\alpha \mathbf{v}\)⁠ are not zero, they are also eigenvectors of ⁠\(A\)⁠ associated with ⁠\(\lambda\)⁠.

The dimension of the eigenspace ⁠\(E\)⁠ associated with ⁠\(\lambda\)⁠, or equivalently the maximum number of linearly independent eigenvectors associated with ⁠\(\lambda\)⁠, is referred to as the eigenvalue's geometric multiplicity and denoted by ⁠\(\gamma_A(\lambda)\)⁠. Because ⁠\(E\)⁠ is also the nullspace of ⁠\(A - \lambda I\)⁠, the geometric multiplicity of ⁠\(\lambda\)⁠ is the dimension of the nullspace of ⁠\(A - \lambda I\)⁠, also called the nullity of ⁠\(A - \lambda I\)⁠. This quantity is related to the size and rank of ⁠\(A - \lambda I\)⁠ by the equation: \[\gamma_A(\lambda) = n - \operatorname{rank}(A - \lambda I).\] 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 ⁠\(n\)⁠. In summary, \[1 \le \gamma_A(\lambda) \le \mu_A(\lambda) \le n.\]

Suppose A has d ≤ n distinct eigenvalues λ₁, ..., λ_d, where the geometric multiplicity of λ_i is γ_A(λ_i). The total geometric multiplicity of A, \[\gamma_A = \sum_{i=1}^d \gamma_A(\lambda_i),\] is the dimension of the sum of all the eigenspaces of A's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of A. By construction, ⁠\(d \le \gamma_A \le n\)⁠. If ⁠\(\gamma_A = n\)⁠, then:

Condensed: the full section is in Wikipedia.

Let A be an arbitrary n × n matrix of complex numbers with eigenvalues λ₁, ..., 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 its eigenvalues: \[\operatorname{tr}(A) = \sum_{i=1}^n a_{ii} = \sum_{i=1}^n \lambda_i = \lambda_1 + \lambda_2 + \cdots + \lambda_n.\]
• The determinant of ⁠\(A\)⁠ is the product of all its eigenvalues: \[\det(A) = \prod_{i=1}^n \lambda_i = \lambda_1 \lambda_2 \cdots \lambda_n.\]
• For any positive integer ⁠\(k\)⁠, the eigenvalues of the ⁠\(k\)⁠th power of ⁠\(A\)⁠, that is, ⁠\(A^k\)⁠, are ⁠\(\lambda_1^k, \ldots, \lambda_n^k\)⁠.
• The eigenvalues of matrix ⁠\(A + I\)⁠ (where ⁠\(I\)⁠ is the identity matrix) are ⁠\(\lambda_1 + 1, \ldots, \lambda_n + 1\)⁠. Moreover, for any ⁠\(\alpha \in \C\)⁠, the eigenvalues of matrix ⁠\(A + \alpha I\)⁠ are ⁠\(\lambda_1 + \alpha, \ldots, \lambda_n + \alpha\)⁠.
• More generally, for any polynomial ⁠\(P\)⁠, the eigenvalues of matrix ⁠\(P(A)\)⁠ are ⁠\(P(\lambda_1), \ldots, P(\lambda_n)\)⁠.
• ⁠\(A\)⁠ is invertible if and only if every eigenvalue ⁠\(\lambda_i\)⁠ is nonzero.
• If ⁠\(A\)⁠ is invertible, then the eigenvalues of ⁠\(A^{-1}\)⁠ are \(\frac{1}{\lambda_1}, \ldots, \frac{1}{\lambda_n}\) and for each pair of corresponding eigenvalues, the geometric multiplicities ⁠\(\gamma_{A}(\lambda_i)\)⁠ and \(\gamma_{A^{-1}}({1 \over \lambda_i})\) coincide. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original up to a scalar factor, for each pair of corresponding eigenvalues, the algebraic multiplicities ⁠\(\mu_{A}(\lambda_i)\)⁠ and \(\mu_{A^{-1}}({1 \over \lambda_i})\) coincide.
• If ⁠\(A\)⁠ is equal to its conjugate transpose ⁠\(A^*\)⁠, that is, ⁠\(A\)⁠ is Hermitian, then every eigenvalue ⁠\(\lambda_i\)⁠ 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 ⁠\(\lambda_i\)⁠ is positive, non-negative, negative, or non-positive, respectively.
• If ⁠\(A\)⁠ is unitary, then every eigenvalue ⁠\(\lambda_i\)⁠ has absolute value ⁠\(\vert \lambda_i \vert = 1\)⁠.

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 multiplies the n × n matrix A in the defining equation, equation (1), \[A \mathbf{v} = \lambda \mathbf{v}.\] The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix A. In this formulation, the defining equation is \[\mathbf{u} A = \kappa \mathbf{u},\] where κ is a scalar and u is a 1 × n matrix. Any row vector u satisfying this equation is called a 'left eigenvector' of A, and κ is still called its associated eigenvalue. Taking the transpose of this equation, \[A^\mathsf{T} \mathbf{u}^\mathsf{T} = \kappa \mathbf{u}^\mathsf{T}.\]

Comparing this equation to equation (1), it follows immediately that a left eigenvector of ⁠\(A\)⁠ is the same as the transpose of a right eigenvector of ⁠\(A^\mathsf{T}\)⁠, with the same eigenvalue. Furthermore, since the characteristic polynomial of ⁠\(A^\mathsf{T}\)⁠ is the same as the characteristic polynomial of ⁠\(A\)⁠, the left and right eigenvectors of ⁠\(A\)⁠ are associated with the same eigenvalues.

A matrix has the same eigenvalues as its transpose, as can be directly seen as follows. Assume ⁠\(\lambda\)⁠ is an eigenvalue of an ⁠\(n \times n\)⁠ matrix ⁠\(A\)⁠ with eigenvector ⁠\(x\)⁠. Then ⁠\(Ax = \lambda x\)⁠; equivalently, ⁠\((A - \lambda I) x = 0\)⁠.

Thus, the columns of ⁠\((A - \lambda I)\)⁠ are linearly dependent. Equivalently, the rank of the matrix is less than ⁠\(n\)⁠.

But as column rank = row rank, the rows are also linearly dependent. Hence, there are numbers ⁠\(y_1, y_2, \ldots, y_n\)⁠, not all zero, such that \(\sum_{i=1}^{i=n} y_i R_i = 0,\) where the ⁠\(R_i\)⁠'s are the rows of ⁠\((A - \lambda I)\)⁠. Let the row vector ⁠\(y = (y_1, y_2, \ldots, y_n)\)⁠; then ⁠\(y \, (A - \lambda I) = 0\)⁠. Taking the transpose, ⁠\((A^\mathsf{T} - \lambda I) \, y^\mathsf{T} = 0\)⁠. Moreover, ⁠\(y^\mathsf{T}\)⁠ is not the zero vector; so ⁠\(\lambda\)⁠ is also an eigenvalue of ⁠\(A^\mathsf{T}\)⁠.

Furthermore, this argument shows that the eigenvalues of \(A\) and \(A^\mathsf{T}\) have the same geometric multiplicity (since column nullity = row nullity).

Suppose the eigenvectors of A form a basis of ⁠\(\C^n\)⁠, or equivalently A has n linearly independent eigenvectors v₁, v₂, ..., v_n (with associated eigenvalues λ₁, λ₂, ..., λ_n). The eigenvectors need not be orthogonal to one another, and the eigenvalues need not be distinct. Define the square matrix Q whose columns are the n linearly independent eigenvectors of A, \[Q = \begin{bmatrix} \mathbf v_1 & \mathbf v_2 & \cdots & \mathbf v_n \end{bmatrix}.\] Since each column of Q is an eigenvector of A, right multiplying A by Q scales each column of Q by its associated eigenvalue: \[AQ = \begin{bmatrix} \lambda_1 \mathbf v_1 & \lambda_2 \mathbf v_2 & \cdots & \lambda_n \mathbf v_n \end{bmatrix}.\]

With this in mind, define the diagonal matrix Λ where each diagonal element Λ_ii is the eigenvalue associated with the ith column of Q. Then \[AQ = Q \varLambda.\] Because the columns of Q are linearly independent, Q is invertible. Right multiplying both sides of the equation by Q, \[A = Q \varLambda Q^{-1};\] or instead left multiplying both sides by Q, \[Q^{-1}AQ = \varLambda.\] 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; 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 PAP is some diagonal matrix D. Left multiplying both by P yields 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.

In conclusion, the eigenvectors of ⁠\(A\)⁠ form a basis of ⁠\(\C^n\)⁠ 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.

In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of H is the maximum value of the quadratic form xHx/xx. A value of x that realizes that maximum is an eigenvector.