Write down a real matrix that has (a) eigenvalues -1,3 and...

Write down a real matrix that has (a) eigenvalues $-1,3$ and corresponding eigenvectors $\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{l}1 \\ 1\end{array}\right)$, (b) eigenvalues $0,2,-2$ and associated eigenvectors $\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 3\end{array}\right)$; (c) an eigenvalue of 3 and corresponding eigenvectors $\left(\begin{array}{r}2 \\ -3\end{array}\right),\left(\begin{array}{l}1 \\ 2\end{array}\right)$; (d) an eigenvalue $-1+2 \mathrm{i}$ and corresponding eigenvector $\left(\begin{array}{c}1+\mathrm{i} \\ 3 \mathrm{i}\end{array}\right)$; (e) an eigenvalue -2 and corresponding eigenvector $\left(\begin{array}{r}2 \\ 0 \\ -1\end{array}\right)$; (f) an eigenvalue $3+\mathrm{i}$ and corresponding eigenvector $\left(\begin{array}{c}1 \\ 2 \mathrm{i} \\ -1-\mathrm{i}\end{array}\right)$.

Write down a real matrix that has
(a) eigenvalues $-1,3$ and corresponding eigenvectors $\left(\begin{array}{r}-1 \\ 2\end{array}\right),\left(\begin{array}{l}1 \\ 1\end{array}\right)$,
(b) eigenvalues $0,2,-2$ and associated eigenvectors $\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 3\end{array}\right)$;
(c) an eigenvalue of 3 and corresponding eigenvectors $\left(\begin{array}{r}2 \\ -3\end{array}\right),\left(\begin{array}{l}1 \\ 2\end{array}\right)$;
(d) an eigenvalue $-1+2 \mathrm{i}$ and corresponding eigenvector $\left(\begin{array}{c}1+\mathrm{i} \\ 3 \mathrm{i}\end{array}\right)$;
(e) an eigenvalue -2 and corresponding eigenvector $\left(\begin{array}{r}2 \\ 0 \\ -1\end{array}\right)$;
(f) an eigenvalue $3+\mathrm{i}$ and corresponding eigenvector $\left(\begin{array}{c}1 \\ 2 \mathrm{i} \\ -1-\mathrm{i}\end{array}\right)$.

Key Concepts

Eigenvalues

Eigenvalues are scalars associated with a square matrix that indicate how the matrix scales vectors in particular directions. They are found by solving the characteristic equation and play a central role in determining the structure and behavior of the transformation represented by the matrix.

Eigenvectors

Eigenvectors are the nonzero vectors that, when transformed by a matrix, result in a scalar multiple of themselves (the corresponding eigenvalue). They point in the directions that remain invariant under the linear transformation, thus providing insight into the matrix’s geometric properties.

Matrix Diagonalization

Matrix diagonalization involves expressing a matrix as PDP?¹, where D is a diagonal matrix containing the eigenvalues and P is a matrix whose columns are the corresponding eigenvectors. This process simplifies computations and analysis by reducing the matrix to a form where its action is simply scaling along independent directions.

Similarity Transformations

Similarity transformations are techniques used to change the basis of a matrix, resulting in an equivalent matrix with the same eigenvalues. They provide a powerful method for analyzing a matrix by transforming it into a simpler or canonical form, such as a diagonal matrix, while preserving its inherent properties.

Complex Eigenvalues in Real Matrices

In real matrices, if there is a complex eigenvalue, its complex conjugate must also be an eigenvalue. This occurs because the characteristic polynomial has real coefficients, leading to complex eigenvalues appearing in conjugate pairs. This concept is important when dealing with real matrices that exhibit rotational or oscillatory behavior.

Multiplicity and Geometric Multiplicity

Multiplicity refers to the number of times an eigenvalue appears as a solution to the characteristic polynomial (algebraic multiplicity), while geometric multiplicity is the number of linearly independent eigenvectors associated with that eigenvalue. Understanding these multiplicities is essential for determining whether a matrix is diagonalizable and for grasping the structure of the eigenspaces.

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