Recent questions in Matrix transformations

Matrix transformations
Answered

Audrina Jackson
2022-07-08

${\mathit{M}}_{0}=\alpha \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$ and ${\mathit{M}}_{1}=\beta \left(\begin{array}{cccc}0& \gamma & 0& -{\gamma}^{\ast}\\ {\gamma}^{\ast}& 0& -{\gamma}^{\ast}& 0\\ 0& \gamma & 0& -{\gamma}^{\ast}\\ \gamma & 0& -\gamma & 0\end{array}\right)$

where $\alpha $ and $\beta $ are constants and $\gamma ={\gamma}_{x}+i{\gamma}_{y}$ is complex.

Is it possible to unitary transform $\mathit{M}$ into block off-diagonal form ${\mathit{M}}_{B}$?

Namely, I want to find a unitary transform $\mathit{U}$ so that I can write down ${\mathit{M}}_{B}=\mathit{U}\mathit{M}{\mathit{U}}^{\ast}$ (here ${\mathit{U}}^{\ast}$ is the conjugate transpose).

Explicitly, the required block off-diagonal matrix is (in general form)

${\mathit{M}}_{B}=\left(\begin{array}{cc}0& \mathit{Q}\\ {\mathit{Q}}^{\ast}& 0\end{array}\right)$ where $\mathit{Q}=\left(\begin{array}{cc}{Q}_{z}& {Q}_{x}-i{Q}_{y}\\ {Q}_{x}+i{Q}_{y}& -{Q}_{z}\end{array}\right)$

Is there a general recipe to find such a unitary transformation matrix $\mathit{U}$ which leads to the block off-diagonal form, $\mathit{M}\to {\mathit{M}}_{B}$?

Matrix transformations
Answered

orlovskihmw
2022-07-07

$\begin{array}{rl}{x}^{\prime}& =4x+2y+14\\ {y}^{\prime}& =2x+7y+42\end{array}$

Find the coordinates of the invariant point of $\mathbf{\text{T}}$. Hence express $\mathbf{\text{T}}$ in the form

$\left(\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}+k\end{array}\right)=\mathbf{\text{A}}\left(\begin{array}{c}x\\ y+k\end{array}\right)$

where $k$ is a positive integer and $\mathbf{\text{A}}$ is a $2\times 2$ matrix.

Matrix transformations
Answered

sebadillab0
2022-07-07

$A=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& ...& {a}_{1n}\\ {a}_{21}& {a}_{22}& ...& {a}_{2n}\\ .& .& .\\ .& .& .\\ .& .& .\\ {a}_{n1}& {a}_{n2}& ...& {a}_{n3}\end{array}\right]$

The basis vectors of $V$ are ${v}_{1},{v}_{2},{v}_{3}...,{v}_{n}$ which are all non standard vectors and similarly ${w}_{1},{w}_{2},...,{w}_{m}$

My question is, in the absence of the basis vectors being standard vectors what is the procedure of finding $T$

Matrix transformations
Answered

aangenaamyj
2022-07-07

Also, say if $T$ is one-one, does this mean it is a matrix transformation and hence a linear transformation?

Matrix transformations
Answered

tripes3h
2022-07-07

If B is a basis of V, then the matrix representation of $[T{]}_{B}^{B}=[{I}_{n}]$.

Let's say C is also a basis of V, then it is clear that

$[T{]}_{C}^{B}\ne [{I}_{n}]$

However, I was taught that matrices representing the same linear transformation in different bases are similar, and the only matrix similar to ${I}_{n}$ is ${I}_{n}$. Thus, $[T{]}_{C}^{B}$ and $[T{]}_{B}^{B}$ are not similar.

Can anyone clear what seems to be a contradiction?

Matrix transformations
Answered

aggierabz2006zw
2022-07-07

$\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$

how can I obtain a matrix 3 * 3 of

$\left[\begin{array}{ccc}a-a& a-b& a-c\\ b-a& b-b& b-c\\ c-a& c-b& c-c\end{array}\right]$

Matrix transformations
Answered

Blericker74
2022-07-06

$y=4x$

The solution should be:

$T=\left(\begin{array}{cc}0.06& 0.235\\ 0.235& 0.94\end{array}\right)$

But somehow i dont know how to get this solution?

Matrix transformations
Answered

pouzdrotf
2022-07-06

Say we have the transformation matrix $A$ of full rank such that ${A}^{-1}\ne {A}^{t},$, i.e., the matrix $A$ consists of linearly independent vectors which aren't orthogonal to each other, a vector v, and the orthonormalized transformation matrix ${A}^{\prime}.$ Is it true that $Av={A}^{\prime}v?$ And if not, is this unimportant?

Matrix transformations
Answered

Jonathan Miles
2022-07-06

$T(x)=\left[\begin{array}{c}-8\\ 9\\ 2\end{array}\right]$

Gven matrix:

$A=\left[\begin{array}{ccc}1& 3& 1\\ -2& 1& 5\\ 0& 2& 2\end{array}\right]$

also aware that T(x) = Ax. I would like to know the general process for finding what x is when given the output vector and a matrix to be multiplied by the unknown input vector x.

$\left[\begin{array}{cccc}1& 0& -2& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

I have tried putting the augmented matrix in reduced row echelon form above, but I am not sure where to go from there.

Matrix transformations
Answered

Aganippe76
2022-07-06

$\left[\begin{array}{cc}12& 5\\ 5& -12\end{array}\right]$

How can I find its eigenvalues/eigenvectors simply by knowing its a reflection-dilation?

Matrix transformations
Answered

gorgeousgen9487
2022-07-06

$\text{A transformation}T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}\text{is represented by the matrix A.}$

$\text{Find the value of k for which the line}y=2x\text{is mapped onto itself under T.}$

Matrix transformations
Answered

Logan Wyatt
2022-07-05

Matrix transformations
Answered

vasorasy8
2022-07-05

and

$T({e}_{2})=T(0,1)=(-\mathrm{sin}\theta ,\mathrm{cos}\theta )$

and

$A=[T({e}_{1})|T({e}_{2})]=\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$

When I rotate a vector $\left[\begin{array}{c}x\\ y\end{array}\right]$ I get

$\left[\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}\end{array}\right]=\left[\begin{array}{c}x\cdot \mathrm{cos}\theta \phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}y\cdot \mathrm{sin}\theta \\ x\cdot \mathrm{sin}\theta \phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}y\cdot \mathrm{cos}\theta \end{array}\right]$

Correct me if I'm wrong, but I thought that column 1 of A $\left[\begin{array}{c}\mathrm{cos}\theta \\ \mathrm{sin}\theta \end{array}\right]$, holds the 'x' values and column 2 holds the 'y' values. What I'm confused about is why does x' contain both an x component and a y component?

Matrix transformations
Answered

Shea Stuart
2022-07-05

Given two lines ${l}_{1}:y=x-3$ and ${l}_{2}:x=1$ find matrix representation of transformation $f$(in standard base) which switch lines each others and find all invariant lines of $f$.

Matrix transformations
Answered

malalawak44
2022-07-05

Matrix transformations
Answered

Frank Day
2022-07-04

$T(\overrightarrow{0})=\overrightarrow{0}$

Since $T$ is a matrix transformation, then for every $\overrightarrow{x}\in {\mathbb{R}}^{n}$ there exists a unique $m\times n$ matrix $A$ such that

$T(\overrightarrow{x})=A\cdot \overrightarrow{x}$

If we let $\overrightarrow{x}=\overrightarrow{0}$, where this zero vector has dimensions $n\times 1$, then

$T(\overrightarrow{0})=A\cdot \overrightarrow{0}=\overrightarrow{0}$

where the $\overrightarrow{0}$ on the right hand side of the equation is an $m\times 1$ vector.

Hence, every transformation maps the zero vector in ${\mathbb{R}}^{n}$ to the zero vector in ${\mathbb{R}}^{m}$

Are there any problems with the proof?

Matrix transformations
Answered

uplakanimkk
2022-07-03

1) a dilation of factor 3 from the x-axis

2) reflection in the x-axis

Matrix transformations
Answered

prirodnogbk
2022-07-03

My intuition tells me that $AB{A}^{T}$ must be symmetric and positive semi-definite, but what is the mathematical proof for this? (why exactly does the transformation preserve symmetry and why is it that possibly negative eigenvalues in $A$ still result in the transformation to be PSD? Or is my intuition wrong)?

Matrix transformations
Answered

mistergoneo7
2022-07-02

Matrix transformations
Answered

Nylah Hendrix
2022-07-02

Now my questions:

- Ok so this vector is invariant. So what? (in my case for attitude determination algorithm I even less understand what this could give me as useful information)

- how does a simple $4\times 4$ matrix actually represent a transformation?

If you are dealing with linear algebra, the chances are high that you will encounter various questions related to matrix transformation. Turning to matrix transformation examples, you will also encounter various geometric transformations, yet these will always be based on algebraic analysis and calculations. The answers that we have presented to various challenges will help you to compare our solutions with your unique matrix transformation example that deals with linear transformation and mapping. Visual assistance is also included and will be essential to see how these are built with the help of the column vectors.