Solved Problems On Eigenvalues And Eigenvectors Pdf

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7. Eigenvalues and Eigenvectors

Geometrically , an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. If T is a linear transformation from a vector space V over a field F into itself and v is a nonzero vector in V , then v is an eigenvector of T if T v is a scalar multiple of v.

This can be written as. There is a direct correspondence between n -by- n square matrices and linear transformations from an n -dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices , or the language of linear transformations.

If V is finite-dimensional, the above equation is equivalent to . Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations.

The prefix eigen- is adopted from the German word eigen cognate with the English word own for "proper", "characteristic", "own".

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. This condition can be written as the equation.

The Mona Lisa example pictured here 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. Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices. If the linear transformation is expressed in the form of an n by n matrix A , then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication.

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:.

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. 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. 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.

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.

At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today. The first numerical algorithm for computing eigenvalues and eigenvectors appeared in , when Richard von Mises published the power method. One of the most popular methods today, the QR algorithm , was proposed independently by John G.

Francis  and Vera Kublanovskaya  in Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.

Consider n -dimensional vectors that are formed as a list of n scalars, such as the three-dimensional vectors. Now consider the linear transformation of n -dimensional vectors defined by an n by n matrix A ,.

If it occurs that v and w are scalar multiples, that is if. Equation 1 is the eigenvalue equation for the matrix A. This polynomial is called the characteristic polynomial of A. Equation 3 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,. As a brief example, which is described in more detail in the examples section later, consider the matrix.

In this example, the eigenvectors are any nonzero scalar multiples of. 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, 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. 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 instead be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,.

The size of each eigenvalue's algebraic multiplicity is related to the dimension n as. Because the eigenspace E is a linear subspace, it is closed under addition. This can be checked using the distributive property of matrix multiplication.

Similarly, because E is a linear subspace, it is closed under scalar multiplication. This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. 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. The following are properties of this matrix and its eigenvalues:. Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. In this formulation, the defining equation is. Taking the transpose of this equation,.

The eigenvalues need not be distinct. Define a square matrix Q whose columns are the n linearly independent eigenvectors of A ,. Since each column of Q is an eigenvector of A , right multiplying A by Q scales each column of Q by its associated eigenvalue,. Because the columns of Q are linearly independent, Q is invertible. 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. The matrix Q is the change of basis matrix of the similarity transformation. Conversely, suppose a matrix A is diagonalizable. 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. In the Hermitian case, eigenvalues can be given a variational characterization. The figure on the right shows the effect of this transformation on point coordinates in the plane.

Note that the eigenvalues are always real if b and c have the same sign, since the quantity under the radical must be positive. The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of A. Consider the cyclic permutation matrix.

This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. For example,. The two complex eigenvectors also appear in a complex conjugate pair,. 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. Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element.

In the example, the eigenvalues correspond to the eigenvectors,. 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.

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. Geometric multiplicities are defined in a later section. For a Hermitian matrix , the norm squared of the j th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix ,. Eigenvalues and eigenvectors

Methods for the calculation of eigenvalues and eigenvectors Part II. Need homework and test-taking help in Linear Algebra? These articles can help you understand more advanced al. ISBN: Are the following two systems of linear equations equivalent? Chapter 6. Eigenvalues and Eigenvectors. Summary To solve the eigenvalue problem for an n by n matrix, follow these steps: 1. Compute the determinant of A.

7. Eigenvalues and Eigenvectors

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The properties of the eigenvalues and their corresponding eigenvectors are also discussed and used in solving questions. Free Mathematics Tutorials. About the author Download E-mail. In this example the eigenvalues are: a , e and g. Geometrically , an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. If T is a linear transformation from a vector space V over a field F into itself and v is a nonzero vector in V , then v is an eigenvector of T if T v is a scalar multiple of v.

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