Transpose of a Matrix and Eigenvalues and Related Questions

Problems and Solutions of Eigenvalue, Eigenvector in Linear Algebra

Problem 12

Let $A$ be an $n \times n$ real matrix. Prove the followings.

(a) The matrix $AA^{\trans}$ is a symmetric matrix.

(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.

(c) The matrix $AA^{\trans}$ is non-negative definite.

(An $n\times n$ matrix $B$ is called non-negative definite if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)

(d) All the eigenvalues of $AA^{\trans}$ is non-negative.

LoadingAdd to solve later

Sponsored Links


Facts.

Before the proofs, we first review several basic properties of the transpose of a matrix.

  1. For any matrices $A$ and $B$ so that the product $AB$ is defined, we have $(AB)^{\trans}=B^{\trans}A^{\trans}$
  2. We have $(A^{\trans})^{\trans}=A$ for any matrix $A$.

Also recall that the eigenvalues of a matrix $A$ are the solutions of the characteristic polynomial $p_A(x)=\det(A-xI)$.

Proof.

(a) The matrix $AA^{\trans}$ is a symmetric matrix.

We compute $(AA^{\trans})^{\trans}=(A^{\trans})^{\trans}A^{\trans}=AA^{\trans}$ and thus $AA^{\trans}$ is a symmetric matrix.

(We used the Fact 1 and 2.)

(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.

We show that the characteristic polynomials of $A$ and $A^{\trans}$ are the same, hence they have exactly same eigenvalues.
Let $p_A(x)$ and $p_{A^{\trans}}(x)$ be the characteristic polynomials of $A$ and $A^{\trans}$, respectively. Then we have
\begin{align*}
p_A(x)=\det(A-xI)=\det(A-xI)^{\trans} =\det(A^{\trans}-xI)=p_{A^{\trans}}(x).
\end{align*}
The first and last equalities are the definition of the characteristic polynomial.
The second equality is true because in general we have $\det(B)=\det(B^{\trans})$ for a square matrix $B$. This completes the proof of (b).

(c) The matrix $AA^{\trans}$ is non-negative definite.

Let $\mathbf{x}$ be an $n$ dimensional vector. Then we have
\begin{align*}
\mathbf{x}^{\trans}AA^{\trans}\mathbf{x}=(A^{\trans}\mathbf{x})^{\trans}(A^{\trans}\mathbf{x})=||A^{\trans}\mathbf{x}|| \geq 0,
\end{align*}
since a norm (length) of a vector is always non-negative.
Thus $AA^{\trans}$ is non-negative definite.

(d) All the eigenvalues of $AA^{\trans}$ is non-negative.

Let $\lambda$ be an eigenvalue of $AA^{\trans}$ and let $\mathbf{x}$ be an eigenvector corresponding to the eigenvalue $\lambda$.
Then we compute
\begin{align*}
\mathbf{x}^{\trans}AA^{\trans}\mathbf{x}=\mathbf{x}^{\trans}\lambda\mathbf{x}=\lambda ||\mathbf{x}||.
\end{align*}
Here the first equality follows from the definitions of the eigenvalue $\lambda$ and eigenvector $\mathbf{x}$. In part (c), we proved that $AA^{\trans}$ is non-negative definite, hence we have $\lambda ||\mathbf{x}|| \geq 0$.
Therefore $\lambda \geq 0$ and this completes the proof of (d).

Related Question.

Problem. 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.

For a solution, see the post ↴
Positive definite real symmetric matrix and its eigenvalues“.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-DefiniteInverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite 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, Linear Algebra Exam Problem)   Proof. (a) Prove that $A$ is […]
  • Positive definite Real Symmetric Matrix and its EigenvaluesPositive definite Real Symmetric Matrix and its Eigenvalues 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 […]
  • Maximize the Dimension of the Null Space of $A-aI$Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
  • Questions About the Trace of a MatrixQuestions About the Trace of a Matrix Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix. (a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of […]
  • Rotation Matrix in Space and its Determinant and EigenvaluesRotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
  • Subspaces of Symmetric, Skew-Symmetric MatricesSubspaces of Symmetric, Skew-Symmetric Matrices Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$. (a) The set $S$ consisting of all $n\times n$ symmetric matrices. (b) The set $T$ consisting of […]
  • Eigenvalues of a Hermitian Matrix are Real NumbersEigenvalues of a Hermitian Matrix are Real Numbers Show that eigenvalues of a Hermitian matrix $A$ are real numbers. (The Ohio State University Linear Algebra Exam Problem)   We give two proofs. These two proofs are essentially the same. The second proof is a bit simpler and concise compared to the first one. […]
  • Symmetric Matrix and Its Eigenvalues, Eigenspaces, and EigenspacesSymmetric Matrix and Its Eigenvalues, Eigenspaces, and Eigenspaces Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix} -1 \\ 2 \\ 0 \\ -1 \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$. Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional. (a) Find an […]

You may also like...

2 Responses

  1. 04/30/2017

    […] For a solution, see the post “Transpose of a matrix and eigenvalues and related questions.“. […]

  2. 05/11/2017

    […] eigenvalues , we deduce that the matrix $A$ has an eigenvalue $1$. (See part (b) of the post “Transpose of a matrix and eigenvalues and related questions.“.) Let $mathbf{x}$ be an eigenvector corresponding to the eigenvalue $1$ (by definition […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Nilpotent Matrix Problems and Solutions
Nilpotent Matrix and Eigenvalues of the Matrix

An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix. Prove the...

Close