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Chapter 8: Problem 14

Suppose that \(A\) and \(A^{\prime}\) are \(n \times n\) matrices. Suppose \(\mathbf{v}\) is an eigenvector of \(A\) associated with the eigenvalue \(\lambda\). Suppose \(\mathbf{v}\) is also an eigenvector of \(A^{\prime}\) associated with the eigenvalue \(\lambda^{\prime}\). Show that \(\mathbf{v}\) is an eigenvector of \(A+A^{\prime}\) associated with \(\lambda+\lambda^{\prime}\).

Short Answer

Expert verified
By rearranging and adding the two equations \(A\mathbf{v} = \lambda\mathbf{v}\) and \(A^{\prime}\mathbf{v} = \lambda^{\prime}\mathbf{v}\), you can prove that \(\mathbf{v}\) is an eigenvector of the matrix \(A + A^{\prime}\) with the associated eigenvalue \(\lambda + \lambda^{\prime}\).

Step by step solution

01

Define the Matrices

In this case, you have two matrices \(A\) and \(A^{\prime}\), both of size \(n \times n\), and an eigenvector \(\mathbf{v}\) that is common to both. For \(A\), the associated eigenvalue is \(\lambda\) and for \(A^{\prime}\), the associated eigenvalue is \(\lambda^{\prime}\).
02

Apply the Eigenvector Property

By definition, if \(\mathbf{v}\) is the eigenvector of a matrix and \(\lambda\) is associated eigenvalue, then the multiplication of the matrix and its eigenvector will yield the product of the eigenvalue and its eigenvector. Thus we have \(A\mathbf{v} = \lambda\mathbf{v}\) and \(A^{\prime}\mathbf{v} = \lambda^{\prime}\mathbf{v}\).
03

Combine the equations

Add the two equations together: \(A\mathbf{v} + A^{\prime}\mathbf{v} = \lambda\mathbf{v} + \lambda^{\prime}\mathbf{v}\)
04

Apply the Matrix Addition

Regroup the equation to express it in terms of matrix addition and scalar multiplication: \((A + A^{\prime})\mathbf{v} = (\lambda + \lambda^{\prime})\mathbf{v}\)
05

Formulate the conclusion

Notice that the last equation shows that \(v\) is an eigenvector of the matrix \(A + A^{\prime}\) with the associated eigenvalue \(\lambda + \lambda^{\prime}\). This is the property you were asked to prove.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Addition
Matrix addition is an operation that takes two matrices of the same dimensions and produces another matrix by adding the corresponding entries. For example, if you have two matrices, A and A', each with the same number of rows and columns (denoted as n 脳 n), their sum is obtained by adding each entry a_ij of matrix A to the corresponding entry a'_ij of matrix A'. The result is a new matrix where each entry is the sum (a_ij + a'_ij).

This principle is essential for understanding how eigenvectors behave under matrix addition. In our example, where the eigenvector v is shared between matrices A and A', we facilitate the proof of their additive properties by aligning the operation of matrix addition with the scalar multiplication of the eigenvector.
Eigenvector Properties
Eigenvectors and eigenvalues are concepts from linear algebra that arise from solving the equation A v = 位 v, where A is a matrix, v is the eigenvector, and 位 is the eigenvalue. One important property is that when a matrix is multiplied by one of its eigenvectors, the result is the same eigenvector multiplied by a scalar (the eigenvalue).

In addition, if a vector is an eigenvector of two matrices separately, then under matrix addition, it is also an eigenvector of the sum of those two matrices with the corresponding eigenvalue being the sum of the two separate eigenvalues. This remains true under the assumption that the matrices involved are of the same size and are combined legally in terms of linear algebra operations.
Linear Algebra Proofs
Linear algebra proofs often require combining several properties and rules to show that a particular statement is true for all cases. In our exercise, the proof that a common eigenvector v of matrices A and A' is also an eigenvector of their sum (A + A'), involves demonstrating that the operations accord with the definitions of matrix multiplication and scalar multiplication.

By following a logical sequence of steps and using known properties of matrices and eigenvectors, we are able to reason and conclude the relationship between these algebraic structures. In essence, proofs in linear algebra often involve such step-by-step logical deductions that must be clearly communicated to validate the underlying theory.
Scalar Multiplication
In linear algebra, scalar multiplication involves multiplying a matrix by a number (known as a scalar) to produce a new matrix. Every entry in the original matrix is multiplied by this scalar. If we have a matrix A and a scalar 位, the result of the scalar multiplication 位础 would be a matrix where each entry a_ij is now 位补冲颈箩.

This concept is deeply intertwined with the properties of eigenvectors. The defining equation A v = 位 v illustrates scalar multiplication on both sides: the matrix A acts upon v, and 位 scales v. These operations underpin many proofs involving eigenvalues and eigenvectors, and our exercise showcases how scalar multiplication helps convey that the eigenvector of a sum of matrices corresponds to the sum of their eigenvalues.

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Most popular questions from this chapter

Suppose \(A\) is the matrix of a linear map \(T: V \rightarrow V\) relative to a basis \(B\) for the finite-dimensional vector space \(V\). a. Prove that if \(v \in V\) is an eigenvector of \(T\) associated with the eigenvalue \(\lambda\), then \([\mathbf{v}]_{B}\) is an eigenvector of \(A\) associated with the eigenvalue \(\lambda\). b. Prove that if \(\mathbf{v} \in \mathbb{R}^{n}\) is an eigenvector of \(A\) associated with the eigenvalue \(\lambda\), then \(L_{B}(\mathbf{v})\) is an eigenvector of \(T\) associated with the eigenvalue \(\lambda\). (Recall that \(L_{B}\) is the linear combination function as defined in Section 6.1.)

a. Generalize Exercise 4 of Section \(8.2\) to show that if two diagonal matrices have the same values occurring on their main diagonals, with each value occurring the same number of times in both matrices but with the values possibly occurring in a different order, then the two matrices are similar. b. Prove that if two diagonal matrices are similar, then they have the same values occurring on their main diagonals, with each value occurring the same number of times in both matrices but with the values possibly occurring in a different order.

a. The position of the hour hand on a clock determines the position of the minute hand. How many times in each twelve-hour period do the hands point in exactly the same direction? How many times do they line up in exactly opposite directions? b. Implement this functional relation between the position of the hands of a clock graphically on a computer or graphing calculator. Allow the user to have some convenient way of specifying the position of the hour hand (perhaps as an angle). Then graph the position of the two hands and give a signal if the hands line up (at least to the resolution of the screen). c. Generalize your program so the relation between the two hands can be any function whose domain and range are \(\mathbb{R}^{2}\). You may wish to normalize the lengths of the domain variable (the hour hand) and the range variable (the minute hand) in order to make them easier to display on the screen. d. Set up your program for a linear relation between the two variables. Try to get the hands to point in the same direction. Verify that the corresponding vectors are eigenvectors of the linear map.

Suppose \(\lambda\) is an eigenvalue of the matrix \(A\). What element of the eigenspace \(E_{\lambda}(\lambda)\) is not an eigenvector of \(A\) associated with \(\lambda\) ?

State and prove simplified versions of the product rule for differentiating matrix functions in the cases where the left or right factor is a constant matrix.
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