Tagged: hermitian matrix

Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix

Problem 585

Consider the Hermitian matrix
1 & i\\
-i& 1

(a) Find the eigenvalues of $A$.

(b) For each eigenvalue of $A$, find the eigenvectors.

(c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix $D$ and a unitary matrix $U$ such that $U^{-1}AU=D$.

Read solution

LoadingAdd to solve later

7 Problems on Skew-Symmetric Matrices

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 $A\mathbf{v}=\mathbf{0}$.

Read solution

LoadingAdd to solve later

Top 10 Popular Math Problems in 2016-2017

It’s been a year since I started this math blog!!

More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).

I made a list of the 10 math problems on this blog that have the most views.

Can you solve all of them?

The level of difficulty among the top 10 problems.
【★★★】 Difficult (Final Exam Level)
【★★☆】 Standard(Midterm Exam Level)
【★☆☆】 Easy (Homework Level)
Read solution

LoadingAdd to solve later

Every Complex Matrix Can Be Written as $A=B+iC$, where $B, C$ are Hermitian Matrices

Problem 425

(a) Prove that each complex $n\times n$ matrix $A$ can be written as
\[A=B+iC,\] where $B$ and $C$ are Hermitian matrices.

(b) Write the complex matrix
i & 6\\
2-i& 1+i
\end{bmatrix}\] as a sum $A=B+iC$, where $B$ and $C$ are Hermitian matrices.

Read solution

LoadingAdd to solve later

Questions About the Trace of a Matrix

Problem 19

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 $B$.

(b) Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.

(c) Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.

Read solution

LoadingAdd to solve later