Problem 6 (a) If $C D=-D C$ (and $D$ i... [FREE SOLUTION]

91影视

Linear Algebra and Its Applications

Gilbert Strang

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 5: Problem 6

(a) If $C D=-D C$ (and $D$ is invertible), show that $C$ is similar to $-C$. (b) Deduce that the eigenvalues of $C$ must come in plus-minus pairs. (c) Show directly that if $C x=\lambda x$, then $C(D x)=-\lambda(D x)$.

Short Answer

Expert verified

Eigenvalues of $ C $ come in plus-minus pairs since $ C $ is similar to $-C$.

Step by step solution

Understand the Problem

We need to show that if $ C D = -D C $ and $ D $ is invertible, then $ C $ is similar to $ -C $. This involves demonstrating that there exists an invertible matrix $ P $ such that $ P^{-1} C P = -C $.

Start with Similarity Transformation

To show similarity, consider $ P = D $. Calculate $ D^{-1} C D $ and show that this results in $ -C $.

Calculate $ D^{-1} C D $

Since $ C D = -D C $, multiply both sides on the left by $ D^{-1} $: $ D^{-1} C D = D^{-1}(-D C) = -C $.

Conclude $ C $ and $ -C $ Similarity

Since $ D^{-1} C D = -C $, $ C $ is similar to $ -C $ via the invertible matrix $ D $.

Deduce Eigenvalue Pair Requirement

If $ C $ is similar to $ -C $, for each eigenvalue $ \lambda $ of $ C $, there exists $ -\lambda $ as an eigenvalue. This is because $ -C $ has eigenvalue $ -\lambda $ if $ C $ has eigenvalue $ \lambda $.

Verify Directly with Eigenvector Transformation

Assume $ C x = \lambda x $ for some eigenvector $ x $. Multiply the equation by $ D $: $ C (D x) = (C D) x = (-D C) x = -\lambda D x $. Thus, $ D x $ is an eigenvector of $ C $ associated with the eigenvalue $-\lambda$.

Confirm Plus-Minus Eigenvalues

The calculation in Step 6 confirms that if $ x $ has eigenvalue $ \lambda $, then $ D x $ has eigenvalue $ -\lambda $, enforcing that eigenvalues occur in pairs of positive and negative.

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Key Concepts

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

Matrix Similarity

Matrix similarity is a concept in linear algebra that refers to the relationship between two matrices that represent the same linear transformation under different bases. Specifically, matrices $ A $ and $ B $ are said to be similar if there is an invertible matrix $ P $ such that $ P^{-1} A P = B $. This means they have the same set of eigenvalues and share many other properties.

One important feature of similar matrices is that they have the same determinant and trace.
If two matrices are similar, they can be seen as different representations of the same operator in different coordinate systems.
In the context of our exercise, proving that $ C $ is similar to $ -C $ involves finding an invertible matrix $ D $ such that $ D^{-1} C D = -C $. This establishes that despite differing by a sign change, $ C $ and $ -C $ are intrinsically related.

Eigenvalues

Eigenvalues are a central concept in linear algebra, representing scalar values associated with a linear transformation that characterizes certain properties of the matrix. When a matrix $ C $ acts on a vector $ x $, it stretches or compresses the vector by some amount, described by an eigenvalue $ \lambda $. The relationship is expressed as $ Cx = \lambda x $.

Eigenvalues are found by solving the characteristic equation $ \det(C - \lambda I) = 0 $, where $ I $ is the identity matrix.
In symmetric matrices, eigenvalues are always real, while in others, they may be complex numbers.
For the exercise at hand, if matrix $ C $ is similar to $ -C $, it ensures that for every eigenvalue $ \lambda $ there exists $ -\lambda $, implying the eigenvalues come in plus-minus pairs. This symmetry results from the transformation behavior of $ C $ and allows the eigenvalues to describe complementary changes in direction or magnitude.

Invertible Matrix

An invertible matrix, also known as a nonsingular matrix, is a matrix that has an inverse. This means there exists another matrix $ D^{-1} $ such that $ DD^{-1} = D^{-1}D = I $, where $ I $ is the identity matrix. The inverse of a matrix can be seen as a matrix that "undoes" the transformation applied by the original matrix.

Inversion is crucial for operations like solving matrix equations and transforming matrices through similarity.
An invertible matrix will always have nonzero eigenvalues, which ties directly to its determinant being non-zero as well.
For our exercise, the matrix $ D $ being invertible is essential because it allows the transformation $ D^{-1} C D = -C $ to hold. This property guarantees that what $ D $ applies is reversible, making it pivotal for establishing similarity between matrices $ C $ and $ -C $. In this way, invertibility facilitates broader transformations and connections between matrices.

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91影视

(a) If \(C D=-D C\) (and \(D\) is invertible), show that \(C\) is similar to \(-C\). (b) Deduce that the eigenvalues of \(C\) must come in plus-minus pairs. (c) Show directly that if \(C x=\lambda x\), then \(C(D x)=-\lambda(D x)\).

Short Answer

Step by step solution

Understand the Problem

Start with Similarity Transformation

Calculate \( D^{-1} C D \)

Conclude \( C \) and \( -C \) Similarity

Deduce Eigenvalue Pair Requirement

Verify Directly with Eigenvector Transformation

Confirm Plus-Minus Eigenvalues

Key Concepts

Matrix Similarity

Eigenvalues

Invertible Matrix

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