7 Problems on Skew-Symmetric Matrices

Linear algebra problems and solutions

Problem 564

Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$.

(a) Prove that $A+B$ is skew-symmetric.

(b) Prove that $cA$ is skew-symmetric for any scalar $c$.

(c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is skew-symmetric.

(d) Suppose that $A$ is real skew-symmetric. Prove that $iA$ is an Hermitian matrix.

(e) Prove that if $AB=-BA$, then $AB$ is a skew-symmetric matrix.

(f) Let $\mathbf{v}$ be an $n$-dimensional column vecotor. Prove that $\mathbf{v}^{\trans}A\mathbf{v}=0$.

(g) Suppose that $A$ is a real skew-symmetric matrix and $A^2\mathbf{v}=\mathbf{0}$ for some vector $\mathbf{v}\in \R^n$. Then prove that $A\mathbf{v}=\mathbf{0}$.

 
LoadingAdd to solve later

Sponsored Links

Proof.

(a) Prove that $A+B$ is skew-symmetric.

We have
\begin{align*}
(A+B)^{\trans}=A^{\trans}+B^{\trans}=(-A)+(-B)=-(A+B).
\end{align*}
Hence $A+B$ is skew-symmetric.

(b) Prove that $cA$ is skew-symmetric for any scalar $c$.

We compute
\begin{align*}
(cA)^{\trans}=cA^{\trans}=c(-A)=-cA.
\end{align*}
Thus, $cA$ is skew-symmetric.

(c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is skew-symmetric.

Using the properties of transpose, we have
\begin{align*}
(P^{\trans}AP)^{\trans}&=P^{\trans}A^{\trans}(P^{\trans})^{\trans}=PA^{\trans}P\\
&=P^{\trans}(-A)P=-(P^{\trans}AP).
\end{align*}
This implies that $P^{\trans}AP$ is skew-symmetric.

(d) Suppose that $A$ is real skew-symmetric. Prove that $iA$ is an Hermitian matrix.

Note that since $A$ is real, we have $\bar{A}=A$.
Then we have
\begin{align*}
(\overline{iA})^{\trans}=(\bar{i}\bar{A})^{\trans}=(-iA)^{\trans}=(-i)A^{\trans}=(-i)(-A)=iA.
\end{align*}
It follows that $iA$ is Hermitian.

(e) Prove that if $AB=-BA$, then $AB$ is a skew-symmetric matrix.

We calculate
\begin{align*}
(AB)^{\trans}&=B^{\trans}A^{\trans}=(-B)(-A)\\
&=BA=-AB,
\end{align*}
where the last step follows from the assumption $AB=-BA$.
This proves that $AB$ is skew-symmetric.

(f) Let $\mathbf{v}$ be an $n$-dimensional column vecotor. Prove that $\mathbf{v}^{\trans}A\mathbf{v}=0$.

Observe that $\mathbf{v}^{\trans}A\mathbf{v}$ is a $1\times 1$ matrix, or just a number.
So we have
\begin{align*}
\mathbf{v}^{\trans}A\mathbf{v}&=(\mathbf{v}^{\trans}A\mathbf{v})^{\trans}=\mathbf{v}^{\trans}A^{\trans}(\mathbf{v}^{\trans})^{\trans}\\
&=\mathbf{v}^{\trans}A^{\trans}\mathbf{v}=\mathbf{v}^{\trans}(-A)\mathbf{v}=-(\mathbf{v}^{\trans}A\mathbf{v}).
\end{align*}
This yields that $2\mathbf{v}^{\trans}A\mathbf{v}=0$, and hence $\mathbf{v}^{\trans}A\mathbf{v}=0$.

(g) Suppose that $A$ is a real skew-symmetric matrix and $A^2\mathbf{v}=\mathbf{0}$ for some vector $\mathbf{v}\in \R^n$. Then prove that $A\mathbf{v}=\mathbf{0}$.

Let us compute the length of the vector $A\mathbf{v}$.
We have
\begin{align*}
\|A\mathbf{v}\|&=(A\mathbf{v})^{\trans}(A\mathbf{v})=\mathbf{v}^{\trans}A^{\trans}A\mathbf{v}\\
&=\mathbf{v}^{\trans}(-A)A\mathbf{v}=-\mathbf{v}^{\trans}A^2\mathbf{v}\\
&=-\mathbf{v}\mathbf{0} &&\text{by assumption}\\
&=0.
\end{align*}
Since the length $\|A\mathbf{v}\|=0$, we conclude that $A\mathbf{v}=\mathbf{0}$.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Express a Hermitian Matrix as a Sum of Real Symmetric Matrix and a Real Skew-Symmetric MatrixExpress 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 written as the sum \[A=B+iC,\] where $B$ is a real symmetric matrix and $C$ is a real skew-symmetric matrix.   Proof. Since […]
  • 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. […]
  • Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All ZeroSum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if \[A_1^2+A_2^2+\cdots+A_m^2=\calO,\] where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.   Hint. Recall that a complex matrix $A$ is Hermitian if […]
  • 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 […]
  • 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 […]
  • If $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is OrthogonalIf $A$ is a Skew-Symmetric Matrix, then $I+A$ is Nonsingular and $(I-A)(I+A)^{-1}$ is Orthogonal 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.   Proof. (a) Prove that the matrices $I-A$ and $I+A$ are nonsingular. The […]
  • Unit Vectors and Idempotent MatricesUnit Vectors and Idempotent Matrices A square matrix $A$ is called idempotent if $A^2=A$. (a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$. Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$. Prove that $P$ is an idempotent matrix. (b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]
  • Symmetric Matrices and the Product of Two MatricesSymmetric Matrices and the Product of Two Matrices Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings. (a) The product $AB$ is symmetric if and only if $AB=BA$. (b) If the product $AB$ is a diagonal matrix, then $AB=BA$.   Hint. A matrix $A$ is called symmetric if $A=A^{\trans}$. In […]

You may also like...

Leave a Reply

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

More in Linear Algebra
Problems and solutions in Linear Algebra
Determine a Condition on $a, b$ so that Vectors are Linearly Dependent

Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\...

Close