Question

The characteristic equation for a second-order matrix $M$ is a quadratic equation. We have, considered in detail the case in which $M$ is a real symmetric matrix and the roots of thecharacteristic equation (eigenvalues) are real, positive, and unequal. Discuss some other possibilities as follows: (a) $M$ real and symmetric, eigenvalues real, one positive and one negative. Show that the plane is reflected in one of the cigenvector lines (as well as stretched or shrunk). Consider as a simple special case $$M=\left(\begin{array}{rr}1& 0\\ 0& -1\end{array}\right)$$ (b) $M$ real and symmetric, eigenvalues equal (and therefore real). Show that $M$ must he a multiple of the unit matrix. Thus show that the deformation consists of dilation or shrinkage in the radial direction (the same in all directions) with no rotation (and reflection in the origin if the root is negative). (c) $M$ real, not symmetric, eigenvalues real and not equal. Show that in this case the eigenvectors are not orthogonal. Hint: Find their dot product. (d) $M$ real, not symmetric, eigenvalues complex, Show that all vectors are rotated, that is, there are no (real) eigenvectors which are unchanged in direction by the transformation. Consider the characteristic equation of a rotation matrix as a special case.

Short answer

In part (a), reflection across an eigenvector line occurs; in part (b), matrix is a scalar multiple of the identity indicating uniform scaling; in part (c), eigenvectors are not orthogonal; and in part (d), complex eigenvalues imply rotation with no fixed directions.

Step-by-step solution

Title - Analyze the case for part (a)

Consider the matrix $$M=\left(\begin{array}{ccc}1& 00& -1\end{array}\right)$$. The eigenvalues are determined by solving the characteristic equation, $$det(M-\lambda I)=0$$. This gives $$det\left(\begin{array}{ccc}1-\lambda & 00& -1-\lambda \end{array}\right)=0$$, leading to the eigenvalues $\lambda =1$ and $\lambda =-1$.

Title - Find eigenvectors for part (a)

For $\lambda =1$, solve $$(M-I)v=0$$:$$\left(\begin{array}{ccc}0& 00& -2\end{array}\right)\left(\begin{array}{c}xy\end{array}\right)=\left(\begin{array}{c}00\end{array}\right)$$, giving the eigenvector $v_1=\left(\begin{array}{c}10\end{array}\right)$. For $\lambda =-1$, solve $$(M+I)v=0$$:$$\left(\begin{array}{ccc}2& 00& 0\end{array}\right)\left(\begin{array}{c}xy\end{array}\right)=\left(\begin{array}{c}00\end{array}\right)$$, producing the eigenvector $v_2=\left(\begin{array}{c}01\end{array}\right)$.

Title - Reflection and stretching for part (a)

The matrix $M$ reflects vectors along the line of eigenvector $v_2$. Any vector $\mathbf{v}=a\left(\begin{array}{c}10\end{array}\right)+b\left(\begin{array}{c}01\end{array}\right)$ is transformed to $M\mathbf{v}=a\left(\begin{array}{c}10\end{array}\right)-b\left(\begin{array}{c}01\end{array}\right)$. This is a reflection in the $x$-axis and changes in magnitude accordingly.

Title - Analyze the case for part (b)

If $M$ is real, symmetric, and has equal eigenvalues, let’s denote $M$ by $$M=cI$$, where $I$ is the identity matrix. Then the deformation by $M$ affects all directions equally.

Title - Show that $M$ is a multiple of the identity matrix for part (b)

Since $M$ satisfies $Mv=\lambda v$ for all $\lambda$, equating to $$\left(\begin{array}{ccc}c& 00& c\end{array}\right)\left(\begin{array}{c}xy\end{array}\right)=c\left(\begin{array}{c}xy\end{array}\right)$$. Thus $M=cI$, indicating uniform scaling (shrink or dilation) without any rotation.

Title - Analyze the case for part (c)

For a real non-symmetric matrix with real unequal eigenvalues, the eigenvectors are not orthogonal. For matrix $$M=\left(\begin{array}{ccc}a& bc& d\end{array}\right)$$, calculate the eigenvalues to determine non-orthogonality.

Title - Determine eigenvalues and eigenvectors for part (c)

Solve the characteristic equation $$det\left(\begin{array}{ccc}a-\lambda & bc& d-\lambda \end{array}\right)=0$$. Once eigenvalues ${\lambda }_{1}eq{\lambda }_2$ are found, compute the dot product of the eigenvectors associated with ${\lambda }_1$ and ${\lambda }_2$.

Title - Analyze the case for part (d)

For a real, non-symmetric matrix with complex eigenvalues, verify the absence of real eigenvectors. Consider a rotation matrix where all vectors undergo rotation.

Title - Characteristic equation for rotation matrix in part (d)

The characteristic equation for a 2D rotation matrix $R$ is $det(R-\lambda I)=0$ and generally has complex roots indicating no fixed direction of vectors.

Key concepts

characteristic equation

The characteristic equation is a polynomial equation derived from a square matrix. It is fundamental to understanding eigenvalues and eigenvectors of a matrix. Given a matrix $M$, the characteristic equation is obtained by solving the determinant of $M-\lambda I=0$, where $I$ is the identity matrix and $\lambda$ represents the eigenvalues.
The roots of this polynomial equation give us the eigenvalues of the matrix. For a second-order matrix $M$, the characteristic equation usually takes the form of a quadratic equation: $$det(M-\lambda I)=0$$.
Finding these eigenvalues is the first step in performing eigenvalue analysis, which helps us in understanding how a matrix transformation affects vectors in a given space.

real symmetric matrix

A real symmetric matrix is a square matrix that is equal to its transpose. Mathematically, for a matrix $M$, this means $M=M^T$. Real symmetric matrices have several important properties:

• All eigenvalues of a real symmetric matrix are real numbers.
• Eigenvectors corresponding to distinct eigenvalues are orthogonal.

These properties simplify the calculation and interpretation of eigenvalues and eigenvectors.
For instance, consider a 2x2 real symmetric matrix:
$$\left(\begin{array}{ccc}a& bb& d\end{array}\right)$$.
This type of matrix often arises in various applications like physics and engineering, where real, symmetric matrices represent scenarios without any complex rotations or skewed transformations.

eigenvectors

Eigenvectors are vectors that only change by a scalar factor when a linear transformation is applied. For a given matrix $M$, an eigenvector $v$ satisfies the equation $Mv=\lambda v$, where $\lambda$ is the corresponding eigenvalue.
In the context of matrix transformations, eigenvectors point in directions that remain invariant except for scaling by the eigenvalue.

• If $\lambda$ is positive, the transformation stretches or dilates the vector.
• If $\lambda$ is negative, it reflects the vector along with stretching.

In the context of a real symmetric matrix, eigenvectors corresponding to different eigenvalues are orthogonal. This property makes it easier to decompose and analyze the effects of matrix transformations on a vector space.

matrix transformation

Matrix transformations describe how vectors change when multiplied by a matrix. This serves as a fundamental concept in linear algebra, playing a critical role in various disciplines such as computer graphics, physics, and engineering.
Depending on the nature of the matrix, transformations can include:

• Stretching/Shrinking: Corresponds to scaling of vectors by eigenvalues.
• Rotation: Occurs when the matrix involves complex eigenvalues.
• Reflection: As seen in the case of certain real symmetric matrices with positive and negative eigenvalues.
• Shearing: Typically represented by non-symmetric matrices.

Each class of matrix transformation adds a different geometric interpretation to how spaces are altered. Understanding these transformations help in visualizing and solving complex problems that involve dynamic systems.