# A Relation of Nonzero Row Vectors and Column Vectors

## Problem 406

Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that
$\mathbf{y}A=\mathbf{y}.$ (Here a row vector means a $1\times n$ matrix.)
Prove that there is a nonzero column vector $\mathbf{x}$ such that
$A\mathbf{x}=\mathbf{x}.$ (Here a column vector means an $n \times 1$ matrix.)

We give two proofs. The first proof does not use the theory of eigenvalues and the second one uses it.

## Proof 1.(Without the theory of eigenvalues)

Let $I$ be the $n\times n$ identity matrix. Then we have
\begin{align*}
\mathbf{0}_{1\times n}=\mathbf{y}A-\mathbf{y}=\mathbf{y}(A-I),
\end{align*}
where $\mathbf{0}_{1\times n}$ is the row zero vector.

Taking the transpose, we have
\begin{align*}
\mathbf{0}_{n\times 1}&=\mathbf{0}_{1\times n}^{\trans}=\left(\,\mathbf{y}(A-I) \,\right)^{\trans}\\
&=(A-I)^{\trans}\mathbf{y}^{\trans}.
\end{align*}

Since the vector $\mathbf{y}$ is nonzero, the transpose $\mathbf{y}^{\trans}$ is a nonzero column vector.
Thus, the above equality yields that the matrix $(A-I)^{\trans}$ is singular.
It follows that the matrix $A-I$ is singular as well.
Hence there exists a nonzero column vector $\mathbf{x}$ such that
$(A-I)\mathbf{x}=\mathbf{0}_{n\times 1},$ and consequently we have
$A\mathbf{x}=\mathbf{x}$ for a nonzero column vector $\mathbf{x}$.

## Proof 2. (Using the theory of eigenvalues)

Taking the conjugate of the both sides of the identity $\mathbf{y}A=\mathbf{y}$, we obtain
$A^{\trans}\mathbf{y}^{\trans}=\mathbf{y}^{\trans}.$ Since $\mathbf{y}$ is a nonzero row vector, $\mathbf{y}^{\trans}$ is a nonzero column vector.
It follows that $1$ is an eigenvalue of the matrix $A^{\trans}$ and $\mathbf{y}^{\trans}$ is a corresponding eigenvector.

Since the matrices $A$ and $A^{\trans}$ has the same 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 $\mathbf{x}$ is nonzero). Then we have
$A\mathbf{x}=\mathbf{x}$ as required.

### More from my site

• Subspaces 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 […]
• Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix? (The Ohio State University, Linear Algebra Final Exam […]
• Compute Determinant of a Matrix Using Linearly Independent Vectors Let $A$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have $A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 […] • Rotation 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 […]
• Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular Find all the values of $x$ so that the following matrix $A$ is a singular matrix. $A=\begin{bmatrix} x & x^2 & 1 \\ 2 &3 &1 \\ 0 & -1 & 1 \end{bmatrix}.$   Hint. Use the fact that a matrix is singular if and only if its determinant is […]
• Eigenvalues of a Matrix and its Transpose are the Same Let $A$ be a square matrix. Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.   Proof. Recall that the eigenvalues of a matrix are roots of its characteristic polynomial. Hence if the matrices $A$ and $A^{\trans}$ […]
• Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose) Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.   Hint. Recall that the rank of a matrix $A$ is the dimension of the range of $A$. The range of $A$ is spanned by the column vectors of the matrix […]
• Find All Values of $x$ so that a Matrix is Singular Let $A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}$ be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.   Hint. Use the fact that a matrix is singular if and only […]

#### You may also like...

##### Express a Hermitian Matrix as a Sum of Real Symmetric Matrix and a Real Skew-Symmetric Matrix

Recall that a complex matrix is called Hermitian if $A^*=A$, where $A^*=\bar{A}^{\trans}$. Prove that every Hermitian matrix $A$ can be...

Close