All Eigenvectors Can Be Extended to Be a Basis

Objectives

1. Acquire the definition of eigenvector and eigenvalue.
2. Learn to find eigenvectors and eigenvalues geometrically.
3. Learn to decide if a number is an eigenvalue of a matrix, and if so, how to detect an associated eigenvector.
4. Recipe: observe a basis for the -eigenspace.
5. Pictures: whether or not a vector is an eigenvector, eigenvectors of standard matrix transformations.
6. Theorem: the expanded invertible matrix theorem.
7. Vocabulary give-and-take: eigenspace.
8. Essential vocabulary words: eigenvector, eigenvalue.

In this section, nosotros define eigenvalues and eigenvectors. These class the about of import facet of the structure theory of square matrices. Equally such, eigenvalues and eigenvectors tend to play a central role in the existent-life applications of linear algebra.

Hither is the well-nigh important definition in this text.

Definition

Allow be an matrix.

1. An eigenvector of is a nonzero vector in such that for some scalar
2. An eigenvalue of is a scalar such that the equation has a nontrivial solution.

If for we say that is the eigenvalue for and that is an eigenvector for

The German prefix "eigen" roughly translates to "self" or "own". An eigenvector of is a vector that is taken to a multiple of itself by the matrix transformation which mayhap explains the terminology. On the other paw, "eigen" is often translated as "feature"; we may call back of an eigenvector as describing an intrinsic, or feature, property of

Eigenvectors are by definition nonzero. Eigenvalues may be equal to nada.

Nosotros do not consider the nada vector to be an eigenvector: since for every scalar the associated eigenvalue would be undefined.

If someone hands y'all a matrix and a vector it is easy to check if is an eigenvector of simply multiply by and run across if is a scalar multiple of On the other hand, given but the matrix information technology is not obvious at all how to find the eigenvectors. We volition acquire how to exercise this in Section five.2.

Case (Verifying eigenvectors)

Example (Verifying eigenvectors)

Example (An eigenvector with eigenvalue )

To say that means that and are collinear with the origin. So, an eigenvector of is a nonzero vector such that and lie on the same line through the origin. In this example, is a scalar multiple of the eigenvalue is the scaling factor.

For matrices that arise equally the standard matrix of a linear transformation, it is often best to draw a picture, then find the eigenvectors and eigenvalues geometrically by studying which vectors are not moved off of their line. For a transformation that is defined geometrically, it is not necessary fifty-fifty to compute its matrix to find the eigenvectors and eigenvalues.

Example (Reflection)

Here is an instance of this. Let be the linear transformation that reflects over the line divers by and let be the matrix for We volition find the eigenvalues and eigenvectors of without doing whatever computations.

This transformation is defined geometrically, so nosotros draw a flick.

The vector is not an eigenvector, considering is not collinear with and the origin.

The vector is not an eigenvector either.

The vector is an eigenvector considering is collinear with and the origin. The vector has the same length every bit simply the contrary direction, so the associated eigenvalue is

The vector is an eigenvector because is collinear with and the origin: indeed, is equal to This means that is an eigenvector with eigenvalue

It appears that all eigenvectors prevarication either on or on the line perpendicular to The vectors on have eigenvalue and the vectors perpendicular to take eigenvalue

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Effigy 8An eigenvector of is a vector such that is collinear with and the origin. Click and drag the head of to convince yourself that all such vectors lie either on or on the line perpendicular to

We will now requite five more than examples of this nature

Example (Projection)

Example (Identity)

Example (Dilation)

Example (Shear)

Example (Rotation)

Here nosotros mention one basic fact near eigenvectors.

Fact (Eigenvectors with distinct eigenvalues are linearly independent)

Let be eigenvectors of a matrix and suppose that the corresponding eigenvalues are distinct (all different from each other). And then is linearly independent.

Proof

Suppose that were linearly dependent. According to the increasing bridge benchmark in Department 2.5, this means that for some the vector is in If we choose the offset such then is linearly independent. Annotation that since

Since is in we can write

for some scalars Multiplying both sides of the above equation by gives

Subtracting times the first equation from the second gives

Since for this is an equation of linear dependence among which is impossible considering those vectors are linearly contained. Therefore, must have been linearly contained later on all.

When this says that if are eigenvectors with eigenvalues then is not a multiple of In fact, any nonzero multiple of is besides an eigenvector with eigenvalue

As a consequence of the above fact, we accept the following.

An matrix has at most eigenvalues.

Suppose that is a square matrix. We already know how to cheque if a given vector is an eigenvector of and in that case to find the eigenvalue. Our side by side goal is to cheque if a given real number is an eigenvalue of and in that example to notice all of the corresponding eigenvectors. Again this will exist straightforward, but more than involved. The only missing slice, then, volition be to discover the eigenvalues of this is the main content of Section v.2.

Let exist an matrix, and let be a scalar. The eigenvectors with eigenvalue if any, are the nonzero solutions of the equation Nosotros can rewrite this equation every bit follows:

Therefore, the eigenvectors of with eigenvalue if any, are the nontrivial solutions of the matrix equation i.e., the nonzero vectors in If this equation has no nontrivial solutions, and then is non an eigenvector of

The above ascertainment is important because it says that finding the eigenvectors for a given eigenvalue means solving a homogeneous system of equations. For instance, if

so an eigenvector with eigenvalue is a nontrivial solution of the matrix equation

This translates to the organisation of equations

This is the same equally the homogeneous matrix equation

i.east.,

Definition

Let be an matrix, and permit be an eigenvalue of The -eigenspace of is the solution set of i.e., the subspace

The -eigenspace is a subspace considering information technology is the null space of a matrix, namely, the matrix This subspace consists of the cypher vector and all eigenvectors of with eigenvalue

Example (Computing eigenspaces)

Example (Computing eigenspaces)

Example (Reflection)

Recipes: Eigenspaces

Let be an matrix and permit be a number.

1. is an eigenvalue of if and just if has a nontrivial solution, if and only if
2. In this example, finding a basis for the -eigenspace of means finding a basis for which tin exist washed by finding the parametric vector form of the solutions of the homogeneous system of equations
3. The dimension of the -eigenspace of is equal to the number of free variables in the system of equations which is the number of columns of without pivots.
4. The eigenvectors with eigenvalue are the nonzero vectors in or equivalently, the nontrivial solutions of

We conclude with an observation well-nigh the -eigenspace of a matrix.

Fact

Let be an matrix.

1. The number is an eigenvalue of if and only if is not invertible.
2. In this case, the -eigenspace of is

Proof

We know that is an eigenvalue of if and but if is nonzero, which is equivalent to the noninvertibility of by the invertible matrix theorem in Section 3.half dozen. In this case, the -eigenspace is by definition

Concretely, an eigenvector with eigenvalue is a nonzero vector such that i.e., such that These are exactly the nonzero vectors in the nix space of

We at present take two new ways of saying that a matrix is invertible, and then nosotros add them to the invertible matrix theorem.

Invertible Matrix Theorem

Allow be an matrix, and allow be the matrix transformation The following statements are equivalent:

1. is invertible.
2. has pivots.
4. The columns of are linearly independent.
5. The columns of span
6. has a unique solution for each in
7. is invertible.
8. is one-to-i.
9. is onto.
11. is not an eigenvalue of

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Source: https://textbooks.math.gatech.edu/ila/eigenvectors.html

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