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.

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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“.


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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 […]

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