Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2
(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
(b) Find all such matrices with rank 2.
Solution.
(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
First we look at the rank 1 case. […]

A Square Root Matrix of a Symmetric Matrix with Non-Negative Eigenvalues
Let $A$ be an $n\times n$ real symmetric matrix whose eigenvalues are all non-negative real numbers.
Show that there is an $n \times n$ real matrix $B$ such that $B^2=A$.
Hint.
Use the fact that a real symmetric matrix is diagonalizable by a real orthogonal matrix.
[…]

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 […]

Symmetric 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 […]

Quiz 13 (Part 1) Diagonalize a Matrix
Let
\[A=\begin{bmatrix}
2 & -1 & -1 \\
-1 &2 &-1 \\
-1 & -1 & 2
\end{bmatrix}.\]
Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.
That is, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that […]

Questions 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 […]

Normal Nilpotent Matrix is Zero Matrix
A complex square ($n\times n$) matrix $A$ is called normal if
\[A^* A=A A^*,\]
where $A^*$ denotes the conjugate transpose of $A$, that is $A^*=\bar{A}^{\trans}$.
A matrix $A$ is said to be nilpotent if there exists a positive integer $k$ such that $A^k$ is the zero […]