For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
votes
1answer
24 views

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of v

Calculate the matrices $[{A{π \over 4}]}^v$ for all possible values of $v$, when $A(\varphi)=\left(\begin{matrix} \cos\varphi &\sin\varphi\\ -\sin\varphi & \cos\varphi\end{matrix}\right)$ . ...
-5
votes
0answers
46 views

Would you please give me your opinion about solving this equation? [closed]

Would you please give me your opinion about solving this equation? [![enter image description here][1]][1] $\sum_{i=0}^{1}\left (-X \right )^{i}*\sum_{J=0}^{7}\sum_{i=0}^{j}\, \gamma _{j}*X^{i}=-T$ ...
3
votes
2answers
74 views

Eigenvalues of the sum of two matrices: one diagonal and the other not.

I'm starting by a simple remark: if $A$ is a $n\times n$ matrix and $\{\lambda_1,\ldots,\lambda_k\}$ are its eigenvalues, then the eigenvalues of matrix $I+A$ (where $I$ is the identity matrix) are ...
-3
votes
2answers
40 views

If $\det(A)=0$, must the null space of $A$ be zero? [closed]

Came along this question: If $\det(A)=0$ for an $N\times N$-dimensional matrix $A$, the null space of $A$ is equal to zero. True or false? Why? Thank you already!
0
votes
7answers
104 views

Is there a matrix with real entries such that $A \ne I_2$ but $A^3 = I_2$.

Is there a matrix with real entries such that $A \ne I_2$ but $A^3 = I_2$. I've actually encountered with this post: $A$ a $n\times n$ matrix with real entries such that $A^3 = I$ but $A \ne I$ ...
-1
votes
1answer
13 views

Augmented matrix [closed]

Animals in an experiment are to be kept under a strict diet. Each animal should receive $20$ grams of protein and $6$ grams of fat. The laboratory technician is able to purchase two food mixes: Mix ...
-2
votes
0answers
33 views

What is ${\sigma _{\varepsilon ,W}}(P)$? [on hold]

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
1
vote
2answers
36 views

Matrix Equation

Imagine the question: If $K$ and $L$ are $2\times 2$ matrices (knowing all of their components) and $KM=L$, solve for the matrix $M$. One simple solution is to set the components of $M$ as $x,y,z,w$ ...
0
votes
2answers
30 views

Equivalence of two different versions of “change of basis matrix”?

I have a question regarding basis change and the matrix that represents it. I understand the concept, though I've noticed a different formula/proof in different math books and I don't understand how ...
1
vote
1answer
30 views

Different formulas for matrix transformations

I am a bit confused about how to get a matrix in a new basis. On the one hand, we always use the multiplication by transformation matrix when we want to receive a matrix in a new basis: $A' = CA$, ...
1
vote
0answers
38 views

How to solve the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ in Matlab when nullitity$(A)\neq 0$

Say, $A= \begin{pmatrix} 1 & 0 &1 \\ 0 & 1 &1 \\ 0 &0 &0 \end{pmatrix}$ and $\overrightarrow{b}= \begin{pmatrix} 8 \\ -5 \\ 0 \end{pmatrix}$ and I want to solve ...
5
votes
5answers
569 views

Why the determinant of a matrix with the sum of each row's elements equal 0 is 0?

I'm trying to understand the proof of a problem, but I'm stuck. In my book they consider that if all lines of a matrix has sum 0 then it's determinant is also 0. I checked some random examples and ...
0
votes
0answers
50 views

Finding permutation matrix

Let $P_{\pi}$ denote a permutation matrix associated to the permutation $\pi:\{1,...,n\}\rightarrow \{1,...,n\}$ and $\sigma$ denote the cyclic permutation $(1 2 ...n)$. If T is the $n\times n$ lower ...
2
votes
2answers
65 views

Conditions for an orthogonal matrix equation

Let $B_1$ and $B_2$ be given $n \times n$ real non-singular matrices and consider the system of equations $$\begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}\begin{bmatrix}B_1 ...
1
vote
1answer
34 views

How to solve this vector equation for optical flow

I am unable to solve for $\textbf{h}$ in the following equation $\sum\limits_{\textbf{x}=1}^n2\partial F(\textbf{x})/\partial\textbf{x}(F(\textbf{x}) + \textbf{h}^{T}\partial F(\textbf{x})/\partial ...
1
vote
1answer
24 views

Replacing pinv with inv in MATLAB

Let $\mathbf{y} = \mathbf{Ax}$ represent a system of equations where $A\in\mathbb{R}^{m\times m}, x\in\mathbb{R}^{m\times 1}$. However rank of $\mathbf{A}$ is $m-1$. I add another equation ...
-1
votes
0answers
24 views

What is subordinate matrix norm?

What is 'subordinate matrix norm' in this question? .
0
votes
0answers
40 views

Wondering how to rotate a normal vector in 4 dimensions?

Saw another post that suggested a answer but need help with the answer and the other post is inactive. I know how to rotate in 3-space using matrix transforms for each axis no problem. Have a very ...
-1
votes
0answers
16 views

Affine Transform Matrix (Rotation) [closed]

Can someone help me with this question? I know how rotation matrix looks like after rotating by y axis but don't know how. Show how to derive the Rotation Matrix about the Y-Axis.
0
votes
0answers
18 views

SVD of partitioned matrix where all cells except one are zero

Let $A$ be a real valued matrix of size $n \times n$. Let the SVD of $A$ be $$A= UDV^T.$$ I am interested in $$Q=VU^T.$$ Now assume we expand $A$ with zero rows and columns to get the block matrix ...
1
vote
1answer
31 views

Non square Matrix multiplication

Assuming we have the following matrix multiplication problem $$ {\bf A x} = {\bf b}$$ and that the dimensions of ${\bf A,x,b}$ are the following $3\times2$, $2\times 1$ and $3\times 1$ How can one ...
1
vote
0answers
28 views

Cayley Hamilton Theorem using LU decompostion

I am trying to find the characterisitic equation of n*n matrix by Cayley Hamilton Theorem using LU Decompostion. Below is my algorithm to find U matrix. ...
1
vote
3answers
70 views

Matrix Exponential and Logarithm

Consider the following matrix $A$: $A = \begin{bmatrix} \cos^2(1) & -\sin(2) & \sin^2(1) \\ \cos(1)\sin(1) & \cos(2) & -\cos(1)\sin(1) \\ \sin^2(1) & \sin(2) & ...
0
votes
1answer
32 views

Permutation and signature matrices “almost commute”

Let $\mathcal{P}$ be the set of all permutation matrices of order $n$ and $\mathcal{S}$ the set of all signature matrices of order $n$. Furthermore, let $$\mathcal{P}\mathcal{S} = \{PS \mid ...
3
votes
1answer
29 views

A = UL factorization [duplicate]

How do I calculate $A=UL$ factorization where $U$ is upper triangular matrix with 1's along the diagonal and $L$ is lower triangular matrix? How is this similar to the $LU$ factorization?
-1
votes
0answers
43 views

What is Heine-Borel theorom

What is Heine-Borel theorom in this link?
1
vote
1answer
15 views

Recurrence Relations for Sequence Counting Hamming Weights

Define $a(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=0, ||x||=k\}|$ and $b(n,k)=|\{x \in \mathbb{F}^{2n}_2 : Hx=1, ||x||=k\}|$ where $||\cdot||$ denotes the Hamming weight of $x$ (i.e. number of non-zero ...
5
votes
2answers
50 views

Why does the Hessian work?

I am working through Susskind's 'The Theoretical Minimum' (on physics) – it also includes some maths. In particular, there is an interlude for which he discusses partial differentiation. He discusses ...
0
votes
4answers
41 views

What is the equation of a 3D line which represents the intersection between two 3D planes?

The intersection defined by the two planes $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$ and $v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$ is a line. What is the equation ...
0
votes
0answers
30 views

How to calculate $det(X^T X)$ efficiently, update one column of X each time

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$. First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...
2
votes
2answers
48 views

Positive semi-definiteness of a matrix whose diagonal elements slightly differ from the sum of the absolute values of other elements in the row

I have a matrix which has the following form: $$ A= \begin{bmatrix} a+b-\varepsilon_1 & -a & -b \\ -a & a+c-\varepsilon_2 & -c \\ -b & -c & b+c-\varepsilon_3 \end{bmatrix} $$ ...
3
votes
1answer
68 views

A matrix as a point in $\mathbb{R}^{nm}$

I just had a really quick question to ask. I was reading a book on linear algebra and have just been trying to wrap my head around what exactly a matrix represents. At one point, the book said "In a ...
3
votes
3answers
48 views

Find $p_{ij}^{(n)}$ for the transition matrix

Let $$P=\begin{bmatrix}\frac{1}{3}&0&\frac{2}{3}\\\frac{1}{3}&\frac{2}{3}&0\\\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}$$ find ...
2
votes
3answers
77 views

Diagonalize a symmetric matrix

let $$A = \left(\begin{array}{cccc} 1&2&3\\2&3&4\\3&4&5 \end{array}\right)$$ I need to find an invertible matrix $P$ such that $P^tAP$ is a diagonal matrix and it's main ...
3
votes
1answer
28 views

Finding $[T_{|W_i}]_{C_i}$

Let $B=\{v_1,v_2,v_3\}$, a basis of $V$ above $\mathbb{R}$. Let $$ [T]_B = \left(\begin{array}{cccc} 6&-3&-2\\4&-1&-2\\10&-5&-3 \end{array}\right)$$ The characteristic ...
0
votes
0answers
8 views

Generating Correlated Samples: Cholesky Decomposition of Correlation Matrix or Covariance Matrix? [duplicate]

I have multiple correlated stochastic processes and I would like to generate correlated samples of them. From my understanding, if I have my samples $Z$ and a Cholesky decomposition of their ...
-6
votes
0answers
47 views

Show that any linear transformation maps the origin to the origin. [closed]

Please solve this question. Show an example to prove it.
1
vote
1answer
75 views

Compactness of a set of matrix polynomials with a norm restriction

Suppose $P_\Delta (\lambda) = (A_m + \Delta _m)\lambda^m + \cdots + (A_1 + \Delta_1)\lambda^1 + (A_0 + \Delta_0)$ is a matrix polynomial, and $\lambda $ is a complex variable. $A_j,\Delta_j \in ...
2
votes
1answer
38 views

Product of projections and commutativity

Let $P_1$, $P_2$, $\dots$, $P_m\in\mathbb{R}^{n\times n}$ be orthogonal projections projecting onto subspaces $V_1$, $V_2$, $\dots$, $V_m$, respectively, and let $P_{1\cap2\cap\dots\cap m}$ denote the ...
0
votes
0answers
33 views

Help with textbook formula

In Bishop - Pattern Recognition and Machine Learning, Section 1, I do not fully understand Formula (1.65). Although it's not stated explicitly, I assume that I is the identity matrix with the ...
-1
votes
1answer
47 views

Column vector of simultaneous equaations' solution

Struggling with some basics of Linear Algebra. Please help. Let's restrict the discussion to 2D space & consider the following simultaneous equations: $2x + 3y = 8, x + 2y = 5$ I understand ...
2
votes
2answers
48 views

Is rank$(AQB)=$rank$(AB)$ if $Q$ is non-singular?

$\newcommand{\rank}{\operatorname{rank}}$We know that $\rank(PA)=\rank(AQ)=\rank(PAQ)=\rank(A)$ where $A\in M_{m\times n}(\mathbb F), P, Q$ are $m\times m, n\times n$ invertible matrices. mean to ...
0
votes
2answers
38 views

Eigenvectors and eigenvalues of matrices

Say that we have a square matrix $M$, and that a non-zero vector $v$ can be an eigenvector of $M$ if $Mv = kv$ for some real number $k$. This real number, $k$, can be called the eigenvalue of $v$ with ...
2
votes
1answer
36 views

What is the number of distinct elements in $S$?

Allow for these values: $$A = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix} \text{ and } B = \begin{pmatrix} 1 ...
2
votes
2answers
17 views

Generate an integer matrix such that all submatrices are non-singular

I need to generate an $\infty \times N$ integer matrix with a few properties. The top $N$ rows (and $N$ columns) should be the identity matrix. Any square submatrix (meaning the result after ...
0
votes
0answers
31 views

Square matrix whose sum of squared elements equals 1.

I'm doing some applied work where I've come across examples that involve real valued square matrices that hold the following property, which expressed using tensor notation is $$A_{ij}A_{ij} = 1$$ ...
0
votes
0answers
14 views

Time derivative of rotation matrix R is the product of a skew matrix and R

Can someone please give a proof that if $R(t)$ is a rotation matrix function of time, then its derivative at time t is equal is equal to a skew matrix times R(t). Thanks
0
votes
0answers
18 views

Reversing a rotation around an offset center of rotation

The best way to generally phrase my question is that I have a sphere offset from its center of rotation and a vector between the sphere and a target object at a known $(\theta,\phi)$ on the sphere. ...
4
votes
1answer
129 views

If $AA^*=AA$, how to prove $A$ is an Hermitian? [duplicate]

If $A$ is an $n \times n$ matrix and $AA^*=AA,$ how to prove $A$ is Hermitian?
0
votes
2answers
52 views

How to Show $M^2=7M-8I$ if $M$ is given in matrix form

$M$ is $2\times 2$ matrix, $m_{11}=3,\ m_{12}=-1,\ m_{21}=-4,\ m_{22}=4$ how to show $M^2=7M-8I$? we can only use substituting or trial and error method or got some more pro method..