what does it mean that eigenvalues equal to zero

Objectives

1. Learn the definition of eigenvector and eigenvalue.
2. Acquire to notice eigenvectors and eigenvalues geometrically.
3. Larn to decide if a number is an eigenvalue of a matrix, and if so, how to observe an associated eigenvector.
4. Recipe: find 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 word: eigenspace.
8. Essential vocabulary words: eigenvector, eigenvalue.

In this section, we define eigenvalues and eigenvectors. These form the most important facet of the structure theory of foursquare matrices. Every bit such, eigenvalues and eigenvectors tend to play a fundamental office in the real-life applications of linear algebra.

Here is the nigh of import definition in this text.

Definition

Let exist 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 "ain". An eigenvector of is a vector that is taken to a multiple of itself by the matrix transformation which perchance explains the terminology. On the other mitt, "eigen" is oft translated every bit "characteristic"; we may call back of an eigenvector as describing an intrinsic, or characteristic, property of

Eigenvectors are by definition nonzero. Eigenvalues may be equal to goose egg.

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

If someone easily you a matrix and a vector information technology is piece of cake to bank check if is an eigenvector of but multiply by and see if is a scalar multiple of On the other hand, given just the matrix it is not obvious at all how to detect the eigenvectors. We volition learn how to do this in Department v.2.

Case (Verifying eigenvectors)

Instance (Verifying eigenvectors)

Example (An eigenvector with eigenvalue )

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

For matrices that arise as the standard matrix of a linear transformation, information technology is often all-time to depict a picture, and so find the eigenvectors and eigenvalues geometrically by studying which vectors are not moved off of their line. For a transformation that is defined geometrically, information technology is not necessary even to compute its matrix to find the eigenvectors and eigenvalues.

Example (Reflection)

Hither is an example of this. Let exist the linear transformation that reflects over the line divers past and let be the matrix for Nosotros will find the eigenvalues and eigenvectors of without doing any computations.

This transformation is defined geometrically, so we draw a picture.

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

The vector is not an eigenvector either.

The vector is an eigenvector because is collinear with and the origin. The vector has the aforementioned length as but the opposite direction, and so the associated eigenvalue is

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

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

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Figure viiiAn eigenvector of is a vector such that is collinear with and the origin. Click and elevate the head of to convince yourself that all such vectors lie either on or on the line perpendicular to

Nosotros volition now give five more examples of this nature

Example (Project)

Example (Identity)

Instance (Dilation)

Example (Shear)

Case (Rotation)

Here we mention 1 basic fact about eigenvectors.

Fact (Eigenvectors with singled-out eigenvalues are linearly independent)

Let be eigenvectors of a matrix and suppose that the respective eigenvalues are singled-out (all different from each other). Then is linearly independent.

Proof

Suppose that were linearly dependent. According to the increasing span criterion in Section two.5, this means that for some the vector is in If nosotros cull the first such so is linearly independent. Note that since

Since is in we can write

for some scalars Multiplying both sides of the above equation past 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 independent after all.

When this says that if are eigenvectors with eigenvalues then is non a multiple of In fact, whatever nonzero multiple of is as well an eigenvector with eigenvalue

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

An matrix has at most eigenvalues.

Suppose that is a square matrix. Nosotros already know how to cheque if a given vector is an eigenvector of and in that case to find the eigenvalue. Our next goal is to check if a given real number is an eigenvalue of and in that case to find all of the corresponding eigenvectors. Once again this will be straightforward, simply more involved. The only missing slice, and so, will be to find the eigenvalues of this is the principal content of Section 5.2.

Let exist an matrix, and let be a scalar. The eigenvectors with eigenvalue if any, are the nonzero solutions of the equation We can rewrite this equation as 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, then is non an eigenvector of

The higher up ascertainment is of import considering it says that finding the eigenvectors for a given eigenvalue means solving a homogeneous arrangement of equations. For instance, if

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

This translates to the system of equations

This is the same equally the homogeneous matrix equation

i.e.,

Definition

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

The -eigenspace is a subspace considering it is the zip space of a matrix, namely, the matrix This subspace consists of the aught vector and all eigenvectors of with eigenvalue

Example (Computing eigenspaces)

Example (Calculating eigenspaces)

Example (Reflection)

Recipes: Eigenspaces

Allow be an matrix and let be a number.

1. is an eigenvalue of if and just if has a nontrivial solution, if and only if
2. In this case, finding a basis for the -eigenspace of means finding a basis for which can exist done 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 gratuitous 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 about 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 example, the -eigenspace of is

Proof

We know that is an eigenvalue of if and only if is nonzero, which is equivalent to the noninvertibility of by the invertible matrix theorem in Section 3.6. 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 aught space of

We at present have ii new ways of maxim that a matrix is invertible, so we add them to the invertible matrix theorem.

Invertible Matrix Theorem

Let 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-one.
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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