Matrix from eigenvalues and eigenvectors

Find a matrix $A \in \mathbb{F}^{3\times 3}$ that has eigenvalues $1, 2, 3$ and eigenvectors the following:

$$\begin{pmatrix}
 1\\
1\\
3
\end{pmatrix}, \;\; \begin{pmatrix}
1\\
1\\
0
\end{pmatrix}, \;\; \begin{pmatrix}
1\\
0\\
0
\end{pmatrix}$$

Solution

Matrix and Cayley Hamilton

Let $A= \begin{pmatrix}
2 &-1  &1 \\
 0&-2  &-1 \\
 0& 3 &2
\end{pmatrix}$.

a) Find the characteristic polynomial of $A$ and deduce that $A$ is invertible.
b) Express $A^{-1}$ as a linear combination of $\mathbb{I}_3, \; A, \; A^2$.
c) Prove that: $A^{2006} - 2A^{2005} = A^2 -2A$.

Solution

On Hermitian matrices

Let $A \in \mathbb{C}^{n\times n}$ be a Hermitian matrix. Prove the following:

a) All the diagonal elements of the matrix $A$ are real.
b) All eigenvalues of $A$ are real.
c) If $\lambda, \mu$ are two dinscint eigenvalues of $A$ and $X, Y$ are their eigenvectors respectively, then $\langle X, Y \rangle=0$.
Solution

Evaluation of determinant

Evaluate the determinant of the $n \times n$ matrix:

$$A=\left({\begin{array}{ccccccc}
1+a^2 & a & 0 & 0 & \cdots & 0 & 0\\
a & 1+a^2 & a & 0 & \cdots & 0 & 0\\
0 & a & 1+a^2 & a & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 &  0 & 0 & \cdots &   a & 0\\
0 & 0 &  0 & 0 & \cdots &  1+a^2 & a\\
0 & 0 &  0 & 0 & \cdots  &   a & 1+a^2
\end{array}}\right)\,,\quad a\in\mathbb{R}$$

Solution

What does a determinant undergo , if ...?

What change does a determinant undergo if to each column (beginning with the second) we add the preceding column and at the same time we add the last to the first?

Solution

Zero matrix

Let $ A\in \mathcal{M}_n\left ( \mathbb{C} \right ) $  with $ n\geq 2 $. If  $ \det \left ( A+X \right )=\det A+\det X $ for every matrix $ X \in \mathcal{M}_n\left ( \mathbb{C} \right ) $ , then prove that $ A=\mathbb{O} $.

Solution

Hankel Determinant

Let $F_n$ denote the $n$-th Fibonacci number. Prove that the determinant of the Hankel matrix defined as:

$$\mathcal{H}=\begin{bmatrix} F_n &F_{n+1}  &F_{n+2}  &\cdots  &F_{2n-1} \\  F_{n+1}&F_{n+2}  & F_{n+3} &\cdots  &F_{2n} \\  F_{n+2}&F_{n+3}  &F_{n+4}  &\cdots  &F_{2n+1} \\  \vdots & \vdots  &\vdots   & \ddots  & \vdots \\  F_{2n-1}&F_{2n}  &F_{2n+1}  &\cdots  & F_{3n-2} \end{bmatrix}$$

is equal to zero.

Solution

Matrices and determinants

Let $n$ be an integer greater or equal to $2$ and let $A, \; B$ be two $n \times n$ real matrices such that:

$$A^{-1} +B^{-1} = (A+B)^{-1}$$

Prove that $\det A = \det B$.

IMC 2015 1st problem

Solution

Square matrices as products of symmetric matrices

Let $\mathbb{F}$ be any field and let $A$ be a square matrix over $\mathbb{F}$. Then $A$ is the product of two symmetric matrices over $\mathbb{F}$.

Solution:

Positive determinant

Let $a_1<a_2<\cdots<a_n$ and $b_1<b_2<\cdots<b_n$ be positive real numbers. Prove that:

$$ \det \begin{pmatrix} e^{a_1 b_1} &e^{a_1 b_2} &\cdots &e^{a_1b_n} \\
e^{a_2 b_1}& e^{a_2 b_2} &\cdots &e^{a_2 b_n} \\
\vdots & \vdots & \ddots &\vdots \\
e^{a_n b_1}&e^{a_n b_2} &\cdots &e^{a_n b_n}
\end{pmatrix}>0$$

Solution

Invertible matrix

Prove that the matrix
$$\begin{bmatrix} 1 & 1 &\cdots  &1 \\  \cos a& \cos 2a &\cdots  &\cos na \\  \cos 2a&\cos 4a  &\cdots  &\cos 2na \\  \vdots & \vdots  & \ddots  &\vdots  \\ \cos(n-1)a &\cos 2(n-1)a  &\cdots  &\cos(n-1)na \end{bmatrix}$$

whereas \( a= \dfrac{2\pi}{n} \) is invertible.

Solution: