Tagged: upper triangular matrix

Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix

Problem 583

Consider the $2\times 2$ complex matrix
a & b-a\\
0& b

(a) Find the eigenvalues of $A$.

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

(c) Diagonalize the matrix $A$.

(d) Using the result of the diagonalization, compute and simplify $A^k$ for each positive integer $k$.

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The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$

Problem 283

Let $F$ be a field and let
\[H(F)=\left\{\, \begin{bmatrix}
1 & a & b \\
0 &1 &c \\
0 & 0 & 1
\end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$.
(The group operation of the Heisenberg group is matrix multiplication.)

Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.

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The Inverse Matrix of an Upper Triangular Matrix with Variables

Problem 275

Let $A$ be the following $3\times 3$ upper triangular matrix.
1 & x & y \\
0 &1 &z \\
0 & 0 & 1
\end{bmatrix},\] where $x, y, z$ are some real numbers.

Determine whether the matrix $A$ is invertible or not. If it is invertible, then find the inverse matrix $A^{-1}$.

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Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent?

Problem 77

A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.

(a) If $A$ is a nilpotent $n \times n$ matrix and $B$ is an $n\times n$ matrix such that $AB=BA$. Show that the product $AB$ is nilpotent.

(b) Let $P$ be an invertible $n \times n$ matrix and let $N$ be a nilpotent $n\times n$ matrix. Is the product $PN$ nilpotent? If so, prove it. If not, give a counterexample.


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Determinant/Trace and Eigenvalues of a Matrix

Problem 9

Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that

(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$

(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$

Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$.

Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues.
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