Find a Matrix that Maps Given Vectors to Given Vectors
Suppose that a real matrix $A$ maps each of the following vectors
\[\mathbf{x}_1=\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}, \mathbf{x}_2=\begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix}, \mathbf{x}_3=\begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix} \]
into the […]
There is at Least One Real Eigenvalue of an Odd Real Matrix
Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.
Prove that the matrix $A$ has at least one real eigenvalue.
We give two proofs.
Proof 1.
Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$.
It is a degree $n$ […]
Cubic Polynomial $x^3-2$ is Irreducible Over the Field $\Q(i)$
Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.
Proof.
Note that the polynomial $x^3-2$ is irreducible over $\Q$ by Eisenstein's criterion (with prime $p=2$).
This implies that if $\alpha$ is any root of $x^3-2$, then the […]
Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite
Suppose $A$ is a positive definite symmetric $n\times n$ matrix.
(a) Prove that $A$ is invertible.
(b) Prove that $A^{-1}$ is symmetric.
(c) Prove that $A^{-1}$ is positive-definite.
(MIT, Linear Algebra Exam Problem)
Proof.
(a) Prove that $A$ is […]
Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$
(a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix}
2 & 0 & -1 \\
0 &2 &-1 \\
-1 & 0 & 1
\end{bmatrix}.\]
(b) Find the inverse matrix of $A$.
Solution
(a) Find a $3\times 3$ nonsingular matrix $A$.
Assume […]
Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
Find a basis for $\Span(S)$ where $S=
\left\{
\begin{bmatrix}
1 \\ 2 \\ 1
\end{bmatrix}
,
\begin{bmatrix}
-1 \\ -2 \\ -1
\end{bmatrix}
,
\begin{bmatrix}
2 \\ 6 \\ -2
\end{bmatrix}
,
\begin{bmatrix}
1 \\ 1 \\ 3
\end{bmatrix}
\right\}$.
Solution.
We […]
Find a Value of a Linear Transformation From $\R^2$ to $\R^3$
Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}$ are unit vectors of $\R^2$ and […]
Companion Matrix for a Polynomial
Consider a polynomial
\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.
Define the matrix
\[A=\begin{bmatrix}
0 & 0 & \dots & 0 &-a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & […]