Construction of a Symmetric Matrix whose Inverse Matrix is Itself

Problem 556

Let $\mathbf{v}$ be a nonzero vector in $\R^n$.
Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$.
Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by
$A=I-a\mathbf{v}\mathbf{v}^{\trans},$ where $I$ is the $n\times n$ identity matrix.

Prove that $A$ is a symmetric matrix and $AA=I$.
Conclude that the inverse matrix is $A^{-1}=A$.

Proof.

$A$ is symmetric

We first show that the matrix $A$ is symmetric.
We calculate using properties of transpose
\begin{align*}
A^{\trans}&=(I-a\mathbf{v}\mathbf{v}^{\trans})^{\trans} && \text{by definition of $A$}\\
&=I^{\trans}-(a\mathbf{v}\mathbf{v}^{\trans})^{\trans}\\
&=I-a(\mathbf{v}^{\trans})^{\trans}\mathbf{v}^{\trans}\\
&=I-a\mathbf{v}\mathbf{v}^{\trans}\\
&=A && \text{by definition of $A$}.
\end{align*}
Hence we have $A^{\trans}=A$, and thus $A$ is symmetric.

$AA=I$ and $A^{-1}=A$

Next, we prove that $AA=I$.

We compute
\begin{align*}
AA&=(I-a\mathbf{v}\mathbf{v}^{\trans})(I-a\mathbf{v}\mathbf{v}^{\trans})\\
&=I(I-a\mathbf{v}\mathbf{v}^{\trans})-a\mathbf{v}\mathbf{v}^{\trans}(I-a\mathbf{v}\mathbf{v}^{\trans})\\
&=I-a\mathbf{v}\mathbf{v}^{\trans}-a\mathbf{v}\mathbf{v}^{\trans}+a^2\mathbf{v}\mathbf{v}^{\trans}\mathbf{v}\mathbf{v}^{\trans}\\
&=I-2a\mathbf{v}\mathbf{v}^{\trans}+a^2\mathbf{v}\mathbf{v}^{\trans}\mathbf{v}\mathbf{v}^{\trans}. \tag{*}
\end{align*}

Note that we have
\begin{align*}
\mathbf{v}\mathbf{v}^{\trans}\mathbf{v}\mathbf{v}^{\trans}&=\mathbf{v}(\mathbf{v}^{\trans}\mathbf{v})\mathbf{v}^{\trans}\\
&=\mathbf{v}\left(\, \frac{2}{a} \,\right)\mathbf{v}^{\trans} &&\text{by definition of $a\neq 0$}\\
&=\frac{2}{a}\mathbf{v}\mathbf{v}^{\trans}.
\end{align*}

Plugging this relation into (*), we obtain
\begin{align*}
AA&=I-2a\mathbf{v}\mathbf{v}^{\trans}+a^2\frac{2}{a}\mathbf{v}\mathbf{v}^{\trans}=I.
\end{align*}
Thus we get $AA=I$.
This implies that the inverse matrix of $A$ is $A$ itself: $A^{-1}=A$.

More from my site

• Find the Inverse Matrix of a Matrix With Fractions Find the inverse matrix of the matrix $A=\begin{bmatrix} \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt] \frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt] -\frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{bmatrix}.$   Hint. You may use the augmented matrix […]
• Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are $\|\mathbf{a}\|=\|\mathbf{b}\|=1$ and the inner product $\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.$ Then determine the length $\|\mathbf{a}-\mathbf{b}\|$. (Note […]
• 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 […]
• Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix 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.   Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
• 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 […]
• The Transpose of a Nonsingular Matrix is Nonsingular Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular.   Definition (Nonsingular Matrix). By definition, $A^{\trans}$ is a nonsingular matrix if the only solution to […]
• Sherman-Woodbery Formula for the Inverse Matrix Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies $\mathbf{v}^{\trans}\mathbf{u}\neq -1.$ Define the matrix […]
• Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let $A, B, C$ be the following $3\times 3$ matrices. \[A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […]

You may also like...

The Range and Null Space of the Zero Transformation of Vector Spaces

Let $U$ and $V$ be vector spaces over a scalar field $\F$. Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for...

Close