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Questions tagged [matrices]

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

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9 views

Null Space of Sum of Two Matrices is a subset or supset of null space of one

Could anyone explain how either of these can be proven? I don't see how either of these statements by themselves can be true, much less how to prove them. $$N(A+B)⊂N(A)$$ $$N(A+B)⊃N(A)$$
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0answers
14 views

Set of Matrices with a given characteristic polynomial is compact [duplicate]

Consider the set of $3\times3$ matrices with the characteristic polynomial $x^3-3x^2+2x-1$. Is the set compact in $M_3(\mathbb{R})\cong \mathbb{R}^9$? The given characteristic polynomial has probably ...
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1answer
22 views

Systems of differential equations

\begin{array} { c } { \text { Solve the initial value problem } } \\ { \mathbf { x } ^ { \prime } = \left( \begin{array} { c c } { 1 } & { - 5 } \\ { 1 } & { - 3 } \end{array} \right) \mathbf {...
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0answers
21 views

Difference between particular solution and general solution?

I have this linear system: \begin{cases} x + 2y - 4z + 5w = 1,\\ x + 3y +5z = 0,\\ y - 4z - 6w = -13. \end{cases} I am asked to find the particular and general solution of this system. I do not know ...
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1answer
33 views

The matrix corresponds to a homomorphism $\mathbb{Z^2} → \mathbb{Z^3}$

In Aluffi's Algebra in 4.3. Reading a presentation it says: For example, take $M=\begin{pmatrix}1&3\\2&3\\5&9\end{pmatrix}$; this matrix corresponds to a homomorphism $\mathbb{Z^2} → \...
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1answer
13 views

What properties of a linear map can be determined from its matrix?

I am currently taking a proofs based linear algebra course for math undergraduates. It's been almost two years since I took a more computational linear algebra course (solving matrix equations, ...
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0answers
8 views

Proving the $(1)$ is a prime element in the set T

I am wondering if there are any logical leaps and anyway for the proof to be concise. Background information: $T=\left\{\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]:\, ...
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0answers
14 views

Determining whether limit goes to inifnity exists for matrices in Jordan canonical form

Recently in class, we covered the Jordan canonical form of matrices. I'm just confused as to when the limit $\lim_n\rightarrow\infty$ exists for certain matrices expressed in such a form - if someone ...
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2answers
32 views

How to find the matrix represented by the polynomials $A^{12}-5A^{11}+…+3I$?

I need to find the characteristic equation of the matrix $ A = \begin{bmatrix} 2&1&1\\ 0&1&0\\ 1&1&2\\ \end{bmatrix} $ and find the matrix represented by $ A^{12}-5A^{11}+7A^{...
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0answers
9 views

Jacobi method- convergence.

I'm trying to find positive-definite matrix $A$ (3x3), such that Jacobi method is not convergent.
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0answers
19 views

Found an article with some equations which seems to be incorrect

I found an article with some equations and got some questions whether some of those equations are correct or not. On the picture you can see that matrix $T_{p_i}$ is on the left side, while matrix $...
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0answers
6 views

Two dimensional Linear Regression Hat Matrix

Let X=(X1, X2)nxp where X1 (nxq) with rank=q and X2 (nx(p-q)) with rank=(p-q). Let H and H1 be hat matrix of X and X1. 1) Prove that HH1=H1 and H1H=H1 2) Prove that (H-H1) is idempotent
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7 views

Given that a symmetric Matrix with only two eigen values and eigen space E_{2}, Is it possible to find the entries in the Matrix

I tried to solve this using linear equations, but then I strongly sense that there is something wrong with the question.
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2answers
18 views

Skew-symmetric non-diagonalizable matrix

Do you have an example of a real skew-symmetric matrix (seen as an operator over $\mathbb{C}^n$) having at least one (purely imaginary) eigenvalue with algebraic multiplicity strictly greater than the ...
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2answers
28 views

True or False ( Invertibility of $A$)

Let $A$ be $10 \times 10$ matrix such that $A^2 + A +I = 0$ then the matrix is invertible. $A^2 + A = –I$ $A(–A–I) = I$ So, – $A$ – $I$ is inverse of A which is clearly $10 \times 10$ matrix, since $...
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1answer
14 views

Show that $L_A$ acts on by orthogonal transformation and in particular rotation.

Let $A$ be a $3\times 3$ orthogonal matrix with determinant $=1$. Let $v$ be an eigen vector corresponding to $1$ of $A$.Let $W=\text{span}\{v\}$. Show that $L_A$ preserves $W^\perp$ and it acts ...
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1answer
22 views

Product of a diagonal matrix and an orthogonal matrix

I am in a situation where I need to prove a property, and I need to know this: Is the product of a diagonal matrix and an orthogonal matrix commutative?
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19 views

calculating stress invariants using matrix

As in the attached picture below, how to find the values of I2 and I3? enter image description here
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1answer
13 views

Mappings to spaces with different numbers of dimensions

The first diagram first question second question I seem to have a good understanding of linear transformation and linear algebra yet I fail to grasp this completely. Question 1) Please explain ...
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0answers
18 views

Transform a common matrix into a form where sum of each rows and columns 1

Is it possible to create a transformation $T$ for a matrix $A$, so a new matrix $C := T(A)$ will have sum for all rows and columns equal to $1$? Can it help if the original matrix $A$ is symmetric?
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1answer
20 views

Doubt about the rank-nullity theorem involving a (block) matrix

Let $P\in\mathbb{R}^{n\times n}$ and $Q\in\mathbb{R}^{n\times m}$ and define $$ S:=[Q, PQ, P^2Q, \ldots, P^{n-1}Q]. $$ Then $S\in\mathbb{R}^{n\times(mn)}$. Now, the book says that, if $\operatorname{...
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0answers
40 views

Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
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0answers
9 views

Relation between Fully Indecomposable Matrix and Doubly Stochastic Matrix

I am studying Combinatoral Matrix Theory by the book written by R. Brualdi. Other problems were well-solved, but I feel hard to solve one problem. The problem is below; Let $A$ be a fully ...
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0answers
46 views

The relation between eigenvalues and inner product of eigenvectors of a real orthogonal matrix where $[R^T]=[R^{-1}]$.

what is the relation between eigenvalues and inner product of eigenvectors of a real orthogonal matrix where $[R^T]=[R^{-1}]$. (Transpose is equal to its Inverse)
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2answers
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Derivative of matrix expression $(Y − A\beta)^TW(Y − A\beta)$ wrt $\beta$.

$Y$ and $\beta$ are $1 \times n$ matrices and $W$ is a diagonal $n \times n$ matrix. What is the best way to think about how to simplify this expression and its derivative to get the expression ...
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1answer
32 views

Is there any easy way to find the determinant of a 4x4 matrix?

I have a 4x4 covariance matrix and want to find the eigenvalues. I know part of the process is to find the determinant: $$\tiny{\begin{align}\begin{vmatrix} 3.33−\lambda & −1.00 & 3.33 & ...
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28 views

What is the meaning of this Matrices/Vectors question & how do I solve it? [on hold]

Give the matrix for the following linear operator: $C = (-10,20,1) \circ (5,-17,-15)$
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1answer
20 views

How do I solve this Matrice/Vectors question? [on hold]

Give the matrix for the following linear operator: $A \vec{x} = (3,2,6) \times \vec{x}$, where $\vec{x}$ is any arbitrary vector.
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14 views

Hash function that describe solid tetromino in the table

I'm trying to find a hash function that describes solid tetromino in a matrix 4x4 consisting of '0' and '1'. Here is what I mean: 1) 1111 0000 0000 0000 - solid tetromino, all this ones have the same ...
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1answer
41 views

Does this property of determinants generalize?

Consider the following $2\times 2$ matrix $$ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}. $$ Let $\Delta = a_{11}a_{22}-a_{21}a_{12}$ be its determinant. Then $A$ ...
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1answer
27 views

Assuming matrices $A,B,C \;\text{and}\;D\;$ are $ n\times n,$ show two ways that $(A+B)(C+D)=AC+AD+BC+BD$

My way first of showing this is by letting $A,B,C \;\text{and}\;D\;$ equal \begin{bmatrix} a_{ij} \end{bmatrix} \begin{bmatrix} b_{ij} \end{bmatrix} \begin{bmatrix} c_{ij} \end{...
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1answer
25 views

Finding the Jordan form and basis for a matrix

I have this matrix $A=\left[ {\begin{array}{ccc} -3&3&-2\\ -7&6&-3\\ 1&-1&2\\ \end{array} } \right]$ I computed the characteristic polynomial $C_A(t)=-(t-2)^2(t-1)$ When I go ...
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0answers
24 views

Show that if $Ax = \lambda x$ for a normal matrix $A,\;$ then $A^{*}x = \bar{\lambda}x$

Suppose $A \in \mathbb{C}^{n\times n}$, and $A$ is normal, show that if $\lambda$ is an eigenvalue of $A$, and $x$ is a corresponding eigenvector of $A$ associated with $\lambda$, then $\bar{\lambda}$ ...
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1answer
97 views

Find eigenvectors of the $(n+1) \times (n+1)$-matrix

Find eigenvectors of the $(n+1) \times (n+1)$-matrix: $$\left(\begin {array}{cccccccc} 0&0&0&0&0&0&-1&0\\ 0&0&0&0&0&-2&0&n\\ 0&0&0&...
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0answers
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I would like an equation of a matrix pseudo-inverse to be explained

I am currently reading a paper titled "A Noise Tolerant Algorithm for Wrist-Mounted Robotic Sensor Calibration with or without Sensor Orientation Measurement", i can email you a snapshot of this paper ...
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1answer
37 views

Questions on symmetric matrices and skew-symmetric matrices

Let $A$ be a $3\times 3\;$ symmetric matrix. Let $U$ be the set of all $3\times 3\;$ skew-symmetric matrices. Let $T : U\to U$ be defined as $T(B)=AB+BA.$ Prove that $T$ is bijective iff the sum of ...
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1answer
22 views

Necessary and sufficient conditions for $x'Ax = 0$

I came across the following problem and I am having a hard time thinking about it. Let $A$ be a $k\times k$ real matrix. Notice that I do not require that $A$ is symmetric, positive definite or ...
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0answers
14 views

Square matrix A as P+iQ where P and Q are hermitian matrix.

This is the link to the answer of the question "Show that every square matrix A can be expressed in one and only one way as P+iQ where P and Q are hermitian matrix" I understand the solution of the ...
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0answers
27 views

Prove that $ \mbox{Tr}(AB)\leq \sum_{i=1}^n \lambda_{i}(A) \lambda_i(B)$, where $A, B$ are $n \times n$ Hermitian matrices

Suppose $A, B$ are $n \times n$ Hermitian matrices, i.e., $ A^{T}=\bar{A}$ and $B^{T}=\bar{B} $, prove that $$ \operatorname{Tr}(AB)\leq \sum_{i=1}^n \lambda_{i}(A)\lambda_i(B), $$ where $ \...
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0answers
15 views

Finding the Upper Bound of the difference between the Inverse of the 2 matrix

Given that $ K = A^{-1} - B^{-1} + A^{-1} - A^{-1}BA^{-1}$, we need to find the upper bound of $K$ where matrix $A = C + I\rho$ and $ B = C_{x} + I\rho $ has dimension $n\times n$, $ C = RDR^{T}$ ...
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0answers
40 views

Eigenvalues of random matrix

I am studying random matrix and stuck by a problem. Is there any way that I can calculate or describe eigenvalues of random matrix? My first attempt was as follows: Let $A$ be random matrix s.t. $A=(...
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1answer
41 views

Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $?

Find a example of $A$ be $4 \times 4$ matrix such that $A$ has rank $2$ but $A^2 =0 $? My attempt : $$A=\begin{bmatrix} 0 & 0 & 1 &0\\0 & 0 & 1 & 0\\0 &0 &...
3
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0answers
46 views

The number of non-singular $n\times n$ matrices over $\mathbb{F}_2$ with exactly $k$ non-zero entries

Suppose $M_{n}^{k}$ is the number of non-singular $n\times n$ matrices over $\mathbb{F}_2$, that have exactly $k$ non-zero entries. Is there some sort of formula to calculate $M_n^k$? If $k < n$ ...
0
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1answer
19 views

Positive/Negative Definite/Semidefinite Test Generality

A test to determine whether a matrix is positive definite, negative definite, positive semidefinite, negative semidefinite, or none of the above, is to calculate the determinant of every cascading ...
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1answer
10 views

What is J in while calculating SST in multiple regression?

I am little confused what actually is the J in the formula of the SST and SSR for multiple regression SST= $Y^T\left[ 1-\frac{1}{n}J\right]Y$ SSR=$Y^T\left[ H-\frac{1}{n}J\right]Y$
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0answers
18 views

Transforming Matrix to Upper Triangular Matrix via Elimination Matrices

Which three matrices E$_{21}$,E$_{31}$,E$_{32}$ put A into triangular form U? A= \begin{bmatrix} 2 & 3 & 1 \\ 4 & -2 & 2 \\ 8 & 1 & 3 \end{bmatrix} and E$_{32}$...
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0answers
18 views

2:1 Diametric rotation matrix for a 2D orthographic projection

I asked this question in the game dev stack exchange, but didn't get a response. This is more of a math question with an application in game development so I hope it's alright if I ask here. I'm ...
1
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1answer
34 views

How to solve for unknown matrix?

How can I solve this? $$ \begin{bmatrix} 1 & 1 \\ 1 & 2 \\ \end{bmatrix} X + \begin{bmatrix} 2 & -1\\ -1 & 1\\ \end{bmatrix} X \begin{bmatrix} 1 & 5 \\ 1 & 2 \\ \...
1
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1answer
24 views

Show that there is an orthogonal matrix $O$ such that $OA_1=A_2O$.

Let $A_1,A_2$ be two real $n \times n$ matrix. And suppose that they are two orthogonal and anti-symmetric matrices. Show that there is an orthogonal matrix $O$ such that $OA_1=A_2O$. I have no idea ...
0
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0answers
62 views

Finding Jordan basis of $5 \times 5$ nilpotent matrix

I have $5 \times 5$ real matrix, which is nilpotent: $$ A = \begin{bmatrix} -2 & 2 & 1 & 3 & -1 \\ 3 & -8 & -2 & -9 & 3 \\ -2 &-8&0 & -6 & 2 \\ -4 &...