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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2
votes
1answer
26 views

Markov chain ergodicity

$(X_n)_n$ is a discrete-time, time-homogenous Markov chain. I have have the following transition matrix and want to show whether the chain is ergodic. $$P = \begin{pmatrix} \frac{1}{2} & 0 & ...
2
votes
1answer
32 views

Can the terms in a 3x3 determinant be any six nonzero numbers?

Given six nonzero real numbers $x_1,\ldots x_6$, can you construct a 3x3 matrix such that the six diagonal products that appear in the determinant are $x_1,\ldots,x_6$, respectively? In other words, ...
0
votes
1answer
88 views

A faster way to tell if a matrix is not non-singular. [closed]

If an n by n square matrix 'W' has an r by r sub-matrix that is singular and (n-1) > r > n/2 when is it true the whole matrix is also singular? Maybe this could in some cases show a matrix is ...
0
votes
1answer
18 views

How to work out the boolean matrix of a relation $S$?

I just wanted a little bit of guidance on how to work out this question in finding the boolean matrix of a relation : Consider the following Hasse diagram of a partial ordering relation $S$ on the ...
5
votes
5answers
484 views

How to compute the determinant of a tridiagonal matrix with constant diagonals?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & ...
12
votes
4answers
122 views

Matrices such that $M^2+M^T=I_n$ are invertible

Let $M$ be an $n\times n$ real matrix such that $M^2+M^T=I_n$. Prove that $M$ is invertible Here is my progress: Playing with determinant: one has $\det(M^2)=\det(I_n-M^T)$ hence ...
0
votes
2answers
27 views

Proof: If $ A \iota_N=0\Rightarrow A^{+'}\iota_N=0$

I want to show that if $ A \iota_N=0\Rightarrow A^{+'}\iota_N=0$ where $A^+$ is the Pseudoinverse.
3
votes
2answers
2k views

Gradient of squared Frobenius norm

I'd like to find the gradient of $\frac{1}{2} \big \| X A^T \big \|_F^2$ with respect to $X_{ij}$. Going by the chain rule in the Matrix Cookbook (eqn 126), it's something like $\partial \, ...
0
votes
3answers
22 views

How do I find a dual basis given the following basis?

$V = \Bbb{R}^3$ and has basis $\mathcal{B} = \{\vec{e_1}-\vec{e_2},\vec{e_1}+\vec{e_2},\vec{e_3}\}$ How do I find the dual basis? This is not homework, but an example that I am struggling to grasp. ...
0
votes
1answer
30 views

Conditional Inverse: Find set of solutions for $AXA=A, A:=\left(1_N-\alpha\iota_N '\right)$.

Given $\alpha \in\mathbb{R}^{N}$ with $\alpha'\iota_N=1$, how can I characterize the set of conditional inverses (or c-inverse) of $\left(1_N-\alpha\iota_N '\right)$ defined as: $\{K\in\mathbb{R}^{N ...
0
votes
0answers
25 views

How to get the unique generalized inverse matrix that we need?

For matrix equation Ax = b (A is a $3×4$ matrix, x is $4×1$ vector , b is $3×1$ vector). now, we have matrix A , vector b and already know that the third value of x is zero. How can we get the vector ...
0
votes
3answers
37 views

Prove that determinant of an odd dimension anti-symmetric matrix is zero

Suppose $A$ is an $(2n+1) \times (2n+1)$ anti-symmetric matrix $(A=-A^T)$. Show that $\det(A)=0$ using Pfaffian formula. Well, in the wiki page, the formula is only defined for matrix with even ...
-4
votes
1answer
35 views

$A$ and $B$ have the same singular values.Why $A$ and $B$ are unitary equivalent? [closed]

Let $A,B \in {M_n}$ and they have the same singular values.Why $A$ and $B$ are unitary equivalent?(by SVD)
1
vote
6answers
1k views

If $A^2+2A+I_n=O_n$ then $A$ is invertible

Let $A$ a matrix of $n\times n$ and $I_n, O_n$ the identity and nule matrix respectively. How to prove that if $A^2+2A+I_n=O_n$ then $A$ is invertible?
0
votes
2answers
52 views

Compute power of a matrix $A$ as $n\rightarrow \infty$

We are given $A^p=A ...A$(p times) And we are given matrix A: $A=\begin{vmatrix}0.6&-0.4&0\\-0.4&0.6&0\\0&0&0.5\end{vmatrix}$ I need to compute $A^p$ as p approach Infinity. ...
0
votes
1answer
25 views

Computing the inverse of linear transformations using matrices

For each of the following linear transformations T, determine whether T is invertible, and compute T-1 if it exists. (a) T: P2(R) $\to$ P2(R) defined by T(f(x)) = f ''(x) + 2 f '(x) - f(x). My ...
8
votes
2answers
322 views

If $ \mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$ then how do we show the $M$ and $N$ have the same eigenvalues?

Let $M,N$ be $n \times n$ square matrices over an algebraically closed field with the properties that the trace of both matrices coincides along with all powers of the matrix. More specifically, ...
0
votes
0answers
34 views

Are members of a diagonal positive definite matrix positive? [duplicate]

If square matrix $M$ is diagonal and positive definite, does it mean that all $m_{ii}$ (diagonal entries) of this matrix are necessarily positive?
0
votes
1answer
36 views

suppose $tr(A^k) = tr(B^k)$ for all $k$=$1,2,…$. why $A$ and $B$ are same characteristic polynomial? . [duplicate]

Let $A,B \in {M_n}$ and suppose $tr(A^k) = tr(B^k)$ for all $k$=$1,2,...$ . Why do $A$ and $B$ possess the same characteristic polynomial?
3
votes
2answers
30 views

Find values of $t$ so that this matrix is positive definite

I will start from this point: $\det{\left(B-\lambda I\right)}=0\Longleftrightarrow\begin{vmatrix}t-\lambda&3&1\\3&t-\lambda&0\\1&0&t-\lambda\end{vmatrix}=0$ Now we will ...
0
votes
0answers
29 views

${A^{ - 1}}$ is similar to $A^*$.why there is a nonsingular $B \in {M_n}$ such that $A=B^{-1}B^*$? . [closed]

Let $A \in {M_n}$ and ${A^{ - 1}}$ is similar to $A^*$.why there is a nonsingular $B \in {M_n}$ such that $A=B^{-1}B^*$?
2
votes
1answer
25 views

Does multiplication by the inverse of a Cholesky matrix preserve order?

Let $n \in \{1, 2, \dots\}$ and let $C \in \mathbb{R}_{n, n}$ be a real, symmetric and positive definite $n \times n$ matrix. Define $B \in \mathbb{R}_{n, n}$ to be the real, lower triangular matrix ...
6
votes
1answer
137 views

Matrices over $\mathbb{Q}[x,y,z]$ which are not equivalent

I have some problems solving the following task: Let $R = \mathbb{Q}[x,y,z]$ and: $$A = \begin{pmatrix} x & y \\ 0 & z \end{pmatrix} \in M_{2,2}(R) \qquad B = \begin{pmatrix} x & 0 ...
0
votes
1answer
39 views

Why : if the matrix $A$ is not invertible, then $L_A$ is not onto.

I was reading a book and the following statement was made: If $A$ is not invertible, then $L_A$ is not onto. Here, the matrix $A$ is $n \times n$ I'm just curious as to why this is true. Thank ...
1
vote
2answers
17 views

Special method of solution for $A\vec x=\vec b$ where $A$ is a square matrix such that $A^tA$ is diagonal and has full rank?

Is there any special shorter method of solution other than cramer's rule for solving a system of $n$ linear equations in $n$ unknowns $A\vec x=\vec b$ where the square matrix $A$ has the property that ...
1
vote
2answers
31 views

Eigenvalues & eigenvectors of a specific product of three matrices

How is possible, without multiplying, find the eigenvalues and eigenvectors of the A matrix ? Which propriety should I use? $$A= \begin{pmatrix} \cos \theta & -\sin \theta\\ \sin \theta & ...
0
votes
1answer
346 views

Simple Probability Matrix

Consider a simple model that predicts whether you pass you next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your ...
2
votes
2answers
87 views

Jordan matrix (de)composition

So any square matrix $A$ can be decomposed into $A = S J S^{-1}$ where $J$ has a normal Jordan form, moreover $A$ and $J$ are similar matrices. My question is quite straightforward. Given arbitrary ...
-1
votes
0answers
24 views

For an $n \times n$ matrix A show that $\lambda$ is an eigenvalue for A [closed]

For an $n \times n$ matrix A show that $\lambda$ is an eigenvalue for A if and only if the determinant $det(A - \lambda I) = 0$ where $I$ is the $n \times n$ identity matrix. Can anyone explain this? ...
0
votes
0answers
30 views

Calculate a matrix at the Power $N$

Knowing that $A^{p}= A * A * \cdots * A$ ($p$ times) that is basically the matrix multiplication propriety of matrix, how can I compute $A^p$ in the limit of $p = \infty$ ?
12
votes
1answer
653 views

Detecting symmetric matrices of the form (low-rank + diagonal matrix)

Let $\Sigma$ be a symmetric positive definite matrix of dimensions $n \times n$. Is there a numerically robust way of checking whether it can be decomposed as $\Sigma = \mathcal{D} + v^t.v$ where $v$ ...
0
votes
2answers
46 views

please how to rotate a matrix $5\times4$ by 45° around the origin $(0,0)$? using matlab

Suppose I have a matrix $M$ of $5\times4$ dimension (this is represent an image) : M = [3 4 8 9; 1 6 7 3; 9 8 3 1; 1 2 2 0; 7 2 3 5]; ...
0
votes
1answer
12 views

Can we ensure convergence for the jacobi method or do we simply trial and error?

For iterative methods for solving systems of equations, we may not always get convergence and it can depend simply on the way in which we write the equations. I understand there are tests which will ...
0
votes
1answer
399 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
1
vote
1answer
28 views

Largest entry in symmetric positive definite matrix

I know why in a symmetric positive definite matrix every entry on the trace is positive entry $a_{ii}>0$. However I don't how to show that the largest value of the matrix is also on it's trace, ...
1
vote
1answer
26 views

Matrix with unit determinant as a product of elementary matrices.

There are three types of elementary matrices: Type 1: matrices obtained by interchanging the ith row of $I$ and jth row of $I$; Type 2: matrices obtained by multiplying the ith row of $I$ by ...
0
votes
1answer
11 views

Change of Coordinates matrix.

If Q is the change of coordinates matrix From some basis B to B', then Q inverse is the change of coordinates matrix from B' to B? Is this true? I think/ know it is the, but don't know how to prove ...
0
votes
1answer
37 views

$n \times n$ matrix Identity Matrix?

Can anyone explain this conceptual problem? If an $n \times n$ matrix is in row reduced echelon form, explain why it is either the identity matrix or else has a row of zeroes? Thanks
2
votes
1answer
151 views

Priority vector and eigenvectors - AHP method

I'm reading about the AHP method (Analytic Hierarchy Process). On page 2 of this document, it says: Given the priorities of the alternatives and given the matrix of preferences for each ...
0
votes
0answers
19 views

Finding basis for image of Linear Transformation

I have been looking up how to find the basis of an image online. I found a solution on StackExchange here It seems that to find the basis we reduce the rows to row echelon form. Then we find the ...
1
vote
1answer
28 views

Invertibility of inertia matrices

The inertia matrix is defined, for a discrete body composed of material points $P_i$ of mass $m_i$ whose position vector $\overrightarrow{CP_i}$ with respect to the centre of mass $C$ has Cartesian ...
1
vote
0answers
17 views

Show that $\{x\in V| \langle x,e \rangle=0 \forall e\in E\} =\{y\in V ~| ~y\perp w_i, 1\leq i \leq k \}$

Let $E$ be subset of a vector space $V$. Let $B =\{w_1,\dots,w_k\}$ be a basis for $E$. Prove: $E^\perp =\{y\in V | y\perp w_i, 1\leq i \leq k \}$ Is my proof correct? Define two sets: (a) ...
4
votes
3answers
203 views

Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.

Let $A$ be an $n \times n$ complex matrix. Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ such that a. A = D + N b. DN = ND and show that these matrices are uniquely ...
0
votes
0answers
10 views

Matrix Decomposition $A=X'X$

Given $A\in\mathbb{R}^{N\times N}$, rank$(A)=N-2$ and $A=A'$. Is there a way to find a matrix $X\in\mathbb{R}^{P \times N}$ such that $A=X'X$? In case this is not possible in general, which additional ...
-4
votes
0answers
20 views

Matrices question [closed]

I need help, this kind of questions are strange to me. Thank you.
1
vote
0answers
24 views

Is there a way to obtain $A^+$ if we know $A$ and $P=AA^+$

Suppose you could obtain $P:=AA^+\in\mathbb{R}^{N\times N}$ experimentally, where $A^+$ is the Moore-Penrose Pseudoinverse. Is there a way to obtain $A^+$? We know that rank$(A)=N-1$.
0
votes
0answers
37 views

${A^{ - 1}}$ is similar to $A^*$.why there is a nonsingular $B \in {M_n}$ such that $A=B^{-1}B^*$? .

Let $A \in {M_n}$ and ${A^{ - 1}}$ is similar to $A^*$.why there is a nonsingular $B \in {M_n}$ such that $A=B^{-1}B^*$?
3
votes
0answers
78 views

Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
1
vote
3answers
47 views

How to find eigenvalues $\lambda>0$ so that matrix A is positive and definite

We are given matrix A: \begin{pmatrix} s & -1 & -1\\ -1 & s & -1\\ -1&-1&s\\ \end{pmatrix} I need to find for which s do A has all eigenvalue $\lambda>0$(positive ...
-1
votes
1answer
52 views

Linear Algebra: Intersection of Subspaces [closed]

Can anyone please help me with this question of my assignment: Let $W$ be a subspace of $V$ (a) Show that there is subspace $U$ of $V$ such that $W\cap U = \{0\}$ and $U+W = V$ (b) Show that there ...