Diagonalization of Matrices

Diagonalization of Matrices

The general procedure of the diagonalization is explained in the post “How to Diagonalize a Matrix. Step by Step Explanation“.

Definition

Let $A, B$ be $n\times n$ matrices.

  1. $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $P^{-1}AP=B$.
  2. $A$ is diagonalizable if there exist a diagonal matrix $D$ and nonsingular matrix $P$ such that $P^{-1}AP=D$. (Namely, if $A$ is diagonalizable if it is similar to a diagonal matrix.)
  3. $A$ is said to be defective if there is an eigenvalue $\lambda$ of $A$ such that the geometric multiplicity of $\lambda$ is less than the algebraic multiplicity of $\lambda$.
Summary

Let $A, B$ be $n\times n$ matrices.

  1. If $A$ and $B$ are similar, then the characteristic polynomials of $A$ and $B$ are the same. Hence the eigenvalues of $A, B$ and their algebraic multiplicities are the same.
  2. $A$ is diagonalizable if and only if $A$ is not defective.
  3. $A$ is diagonalizable if and only if $\R^n$ has an eigenbasis of $A$ (a basis consisting of eigenvectors).
  4. $A$ is diagonalizable if and only if there are $n$ linearly independent eigenvectors of $A$.
  5. If $A$ has $n$ distinct eigenvalues, then $A$ is diagonalizable.
  6. If $\mathbf{v}_1, \dots, \mathbf{v}_n$ are linearly independent eigenvectors of $A$ corresponding to the eigenvalues $\lambda_1, \dots, \lambda_n$ (not necessarily distinct), then $S^{-1}AS=D$, where $S=[\mathbf{v}_1, \dots, \mathbf{v}_n]$ and $D=\diag(\lambda_1, \dots, \lambda_n)$.

=solution

Problems

  1. Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix. Prove the following statements.
    (a) If $A$ is similar to $B$, then $B$ is similar to $A$.
    (b) $A$ is similar to itself.
    (c) If $A$ is similar to $B$ and $B$ is similar to $C$, then $A$ is similar to $C$.
    (d) If $A$ is similar to the identity matrix $I$, then $A=I$.
    (e) If $A$ or $B$ is nonsingular, then $AB$ is similar to $BA$.
    (f) If $A$ is similar to $B$, then $A^k$ is similar to $B^k$ for any positive integer $k$.

  2. Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.
  3. (a) Is the matrix $A=\begin{bmatrix}
    1 & 2\\
    0& 3
    \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
    3 & 0\\
    1& 2
    \end{bmatrix}$?
    (b) Is the matrix $A=\begin{bmatrix}
    0 & 1\\
    5& 3
    \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
    1 & 2\\
    4& 3
    \end{bmatrix}$?
    (c) Is the matrix $A=\begin{bmatrix}
    -1 & 6\\
    -2& 6
    \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
    3 & 0\\
    0& 2
    \end{bmatrix}$?
    (d) Is the matrix $A=\begin{bmatrix}
    -1 & 6\\
    -2& 6
    \end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}
    1 & 2\\
    -1& 4
    \end{bmatrix}$?

  4. Determine whether the matrix $A=\begin{bmatrix}
    1 & 4\\
    2 & 3
    \end{bmatrix}$ is diagonalizable. If so, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
    (The Ohio State University)

  5. Diagonalize the matrix
    \[A=\begin{bmatrix}
    4 & -3 & -3 \\
    3 &-2 &-3 \\
    -1 & 1 & 2
    \end{bmatrix}\] by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

  6. Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix.
    If
    \[P^{-1}AP=\begin{bmatrix}
    1 & 2 & 3 \\
    0 &4 &5 \\
    0 & 0 & 6
    \end{bmatrix},\] then find all the eigenvalues of the matrix $A^2$.

  7. Let $A= \begin{bmatrix}
    1 & 2\\
    2& 1
    \end{bmatrix}$. Compute $A^n$ for any $n \in \N$.

  8. Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.

  9. (a) Is every diagonalizable matrix invertible?
    (b) Is every invertible matrix diagonalizable?

  10. Determine whether the matrix $A=\begin{bmatrix}
    0 & 1 & 0 \\
    -1 &0 &0 \\
    0 & 0 & 2
    \end{bmatrix}$ is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

  11. For which values of constants $a, b$ and $c$ is the matrix $A=\begin{bmatrix}
    7 & a & b \\
    0 &2 &c \\
    0 & 0 & 3
    \end{bmatrix}$ diagonalizable?
    (The Ohio State University)

  12. Let
    \[A=\begin{bmatrix}
    1 & 3 & 3 \\
    -3 &-5 &-3 \\
    3 & 3 & 1
    \end{bmatrix} \text{ and } B=\begin{bmatrix}
    2 & 4 & 3 \\
    -4 &-6 &-3 \\
    3 & 3 & 1
    \end{bmatrix}.\] For this problem, you may use the fact that both matrices have the same characteristic polynomial:
    \[p_A(\lambda)=p_B(\lambda)=-(\lambda-1)(\lambda+2)^2.\] (a) Find all eigenvectors of $A$.
    (b) Find all eigenvectors of $B$.
    (c) Which matrix $A$ or $B$ is diagonalizable?
    (d) Diagonalize the matrix stated in (c), i.e., find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$ or $B=PDP^{-1}$.
    (Stanford University)

  13. Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors $\begin{bmatrix}
    1 \\
    0
    \end{bmatrix}$ and $\begin{bmatrix}
    2 \\
    1
    \end{bmatrix}$, respectively.

  14. Let $A$ and $B$ be $n\times n$ matrices. Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$. Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.
  15. Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues. Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.
    (Stanford University)

  16. Let
    \[A=\begin{bmatrix}
    \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\
    \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\
    \frac{3}{7} & \frac{3}{7} & \frac{1}{7}
    \end{bmatrix}\] be $3 \times 3$ matrix. Find
    \[\lim_{n \to \infty} A^n.\] (Nagoya University Linear Algebra Exam)

  17. Let
    \[\begin{bmatrix}
    0 & 0 & 1 \\
    1 &0 &0 \\
    0 & 1 & 0
    \end{bmatrix}.\] (a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers?
    (b) Calculate $A^{2009}$.
    (Princeton University)

  18. Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix.

  19. Let
    \[A=\begin{bmatrix}
    2 & -1 & -1 \\
    -1 &2 &-1 \\
    -1 & -1 & 2
    \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.

  20. Let $A$ be an $n\times n$ matrix with the characteristic polynomial
    \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable.
    (a) Find the size of the matrix $A$.
    (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue $\lambda=2$.
    (c) Find the nullity of $A$.
    (The Ohio State University)

  21. Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers. Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.

  22. Find a square root of the matrix
    \[A=\begin{bmatrix}
    1 & 3 & -3 \\
    0 &4 &5 \\
    0 & 0 & 9
    \end{bmatrix}.\] How many square roots does this matrix have?
    (University of California, Berkeley)

  23. Suppose the following information is known about a $3\times 3$ matrix $A$.
    \[A\begin{bmatrix}
    1 \\
    2 \\
    1
    \end{bmatrix}=6\begin{bmatrix}
    1 \\
    2 \\
    1
    \end{bmatrix},
    \quad
    A\begin{bmatrix}
    1 \\
    -1 \\
    1
    \end{bmatrix}=3\begin{bmatrix}
    1 \\
    -1 \\
    1
    \end{bmatrix}, \quad
    A\begin{bmatrix}
    2 \\
    -1 \\
    0
    \end{bmatrix}=3\begin{bmatrix}
    1 \\
    -1 \\
    1
    \end{bmatrix}.\] (a) Find the eigenvalues of $A$.
    (b) Find the corresponding eigenspaces.
    (c) Is $A$ a diagonalizable matrix? Is $A$ an invertible matrix? Is $A$ an idempotent matrix?
    (Johns Hopkins University)

  24. Diagonalize the matrix $A=\begin{bmatrix}
    1 & 1 & 1 \\
    1 &1 &1 \\
    1 & 1 & 1
    \end{bmatrix}$. Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
    (The Ohio State University)

  25. Prove that the matrix
    \[A=\begin{bmatrix}
    0 & 1\\
    -1& 0
    \end{bmatrix}\] is diagonalizable.
    Prove, however, that $A$ cannot be diagonalized by a real nonsingular matrix.
    That is, there is no real nonsingular matrix $S$ such that $S^{-1}AS$ is a diagonal matrix.

  26. Let $A=\begin{bmatrix}
    1-a & a\\
    -a& 1+a
    \end{bmatrix}$ be a $2\times 2$ matrix, where $a$ is a complex number. Determine the values of $a$ such that the matrix $A$ is diagonalizable.
    (Nagoya University)

  27. Consider the $2\times 2$ complex matrix $A=\begin{bmatrix}
    a & b-a\\
    0& b
    \end{bmatrix}$.
    (a) Find the eigenvalues of $A$.
    (b) For each eigenvalue of $A$, determine the eigenvectors.
    (c) Diagonalize the matrix $A$.
    (d) Using the result of the diagonalization, compute and simplify $A^k$ for each positive integer $k$.

  28. Consider the complex matrix
    \[A=\begin{bmatrix}
    \sqrt{2}\cos x & i \sin x & 0 \\
    i \sin x &0 &-i \sin x \\
    0 & -i \sin x & -\sqrt{2} \cos x
    \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the matrix $A$ is diagonalizable. When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.

  29. Consider the Hermitian matrix $A=\begin{bmatrix}
    1 & i\\
    -i& 1
    \end{bmatrix}$.
    (a) Find the eigenvalues of $A$.
    (b) For each eigenvalue of $A$, find the eigenvectors.
    (c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.

  30. Let $A$ be an $n\times n$ complex matrix. Let $S$ be an invertible matrix.
    (a) If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.
    (b) If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an eigenvalue of $A$.
    (c) Suppose that all the eigenvalues of $A$ are integers and $\det(A) > 0$. If $n$ is odd and $SAS^{-1}=A^{-1}$, then prove that $1$ is an eigenvalue of $A$.

  31. Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$. Then prove the following statements.
    (a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number.
    (b) The rank of $A$ is even.

  32. Let $A$ be an $n\times n$ real symmetric matrix. Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality $\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2$.
  33. A real symmetric $n \times n$ matrix $A$ is called positive definite if $\mathbf{x}^{\trans}A\mathbf{x}>0$ for all nonzero vectors $\mathbf{x}$ in $\R^n$.
    (a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive.
    (b) Prove that if eigenvalues of a real symmetric matrix $A$ are all positive, then $A$ is positive-definite.

  34. Suppose $A$ is a positive definite symmetric $n\times n$ matrix.
    (a) Prove that $A$ is invertible.
    (b) Prove that $A^{-1}$ is symmetric.
    (c) Prove that $A^{-1}$ is positive-definite.
    (MIT)

  35. Let $A=\begin{bmatrix}
    1 & 2\\
    4& 3
    \end{bmatrix}$.
    (a) Find eigenvalues of the matrix $A$.
    (b) Find eigenvectors for each eigenvalue of $A$.
    (c) Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
    (d) Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.
    (e) Calculate $A^{100}$. (You do not have to compute $5^{100}$.)
    (f) Calculate $(A^3-5A^2+3A+I)^{100}$. Let $w=2^{100}$. Express the solution in terms of $w$.

  36. Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.
  37. Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$. Find all the square roots of the matrix $A=\begin{bmatrix}
    2 & 2\\
    2& 2
    \end{bmatrix}$.

  38. Let $A$ be an $n\times n$ idempotent complex matrix. Then prove that $A$ is diagonalizable.
  39. Let $A$ be an $n\times n$ real skew-symmetric matrix.
    (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular.
    (b) Prove that $B=(I-A)(I+A)^{-1}$ is an orthogonal matrix.

  40. Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
    (a) Prove that for sufficiently small positive real $\epsilon$, the equation $A\mathbf{x}+\epsilon\mathbf{x}=\mathbf{v}$ has a unique solution $\mathbf{x}=\mathbf{x}(\epsilon) \in \R^n$.
    (b) Evaluate $\lim_{\epsilon \to 0^+} \epsilon \mathbf{x}(\epsilon)$ in terms of $\mathbf{v}$, the eigenvectors of $A$, and the inner product $\langle\, ,\,\rangle$ on $\R^n$.
    (University of California)

  41. Let $I$ be the $2\times 2$ identity matrix. Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.
  42. Prove that a positive definite matrix has a unique positive definite square root.