Category: Linear Algebra

The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization

Problem 539

Consider the $2\times 2$ real matrix
\[A=\begin{bmatrix}
1 & 1\\
1& 3
\end{bmatrix}.\]

(a) Prove that the matrix $A$ is positive definite.

(b) Since $A$ is positive definite by part (a), the formula
\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans} A \mathbf{y}\] for $\mathbf{x}, \mathbf{y} \in \R^2$ defines an inner product on $\R^n$.
Consider $\R^2$ as an inner product space with this inner product.

Prove that the unit vectors
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}\] are not orthogonal in the inner product space $\R^2$.

(c) Find an orthogonal basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ of $\R^2$ from the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$ using the Gram-Schmidt orthogonalization process.

 

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A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space

Problem 538

(a) Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix.
Prove that
\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}\] defines an inner product on the vector space $\R^n$.

(b) Let $A$ be an $n\times n$ real matrix. Suppose that
\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}\] defines an inner product on the vector space $\R^n$.

Prove that $A$ is symmetric and positive definite.

 

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Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$

Problem 533

Consider the complex matrix
\[A=\begin{bmatrix}
\sqrt{2}\cos x & i \sin x & 0 \\
i \sin x &0 &-i \sin x \\
0 & -i \sin x & -\sqrt{2} \cos x
\end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$.

Determine for which values of $x$ the matrix $A$ is diagonalizable.
When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.

 

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Is the Linear Transformation Between the Vector Space of 2 by 2 Matrices an Isomorphism?

Problem 528

Let $V$ denote the vector space of all real $2\times 2$ matrices.
Suppose that the linear transformation from $V$ to $V$ is given as below.
\[T(A)=\begin{bmatrix}
2 & 3\\
5 & 7
\end{bmatrix}A-A\begin{bmatrix}
2 & 3\\
5 & 7
\end{bmatrix}.\] Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.

 

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Unit Vectors and Idempotent Matrices

Problem 527

A square matrix $A$ is called idempotent if $A^2=A$.


(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.

Prove that $P$ is an idempotent matrix.


(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.

Prove that $Q$ is an idempotent matrix.


(c) Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).

 

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How to Prove a Matrix is Nonsingular in 10 Seconds

Problem 509

Using the numbers appearing in
\[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\] we construct the matrix \[A=\begin{bmatrix}
3 & 14 &1592& 65358\\
97932& 38462643& 38& 32\\
7950& 2& 8841& 9716\\
939937510& 5820& 974& 9
\end{bmatrix}.\]

Prove that the matrix $A$ is nonsingular.

 

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The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$

Problem 505

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]

Using the formula, calculate the inverse matrix of $\begin{bmatrix}
2 & 1\\
1& 2
\end{bmatrix}$.

 

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10 True of False Problems about Nonsingular / Invertible Matrices

Problem 500

10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.

The quiz is designed to test your understanding of the basic properties of these topics.

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.
Click the View question button to see the solutions.

 

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The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane

Problem 498

Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself of the reflection across a line $y=mx$ for some $m\in \R$.

Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where
\[\mathbf{e}_1=\begin{bmatrix}
1 \\
0
\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}
0 \\
1
\end{bmatrix}.\]

 

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A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal

Problem 492

Let
\[D=\begin{bmatrix}
d_1 & 0 & \dots & 0 \\
0 &d_2 & \dots & 0 \\
\vdots & & \ddots & \vdots \\
0 & 0 & \dots & d_n
\end{bmatrix}\] be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$.
Let $A=(a_{ij})$ be an $n\times n$ matrix such that $A$ commutes with $D$, that is,
\[AD=DA.\] Then prove that $A$ is a diagonal matrix.

 

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