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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-1
votes
0answers
24 views

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

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
22 views

What do the ellipses mean in $x^TW_{\dots ij}$

I cam across this notation: $x^TW_{\dots ij}$. I don't understand the notation. Why are there 3 dots? Source: http://jmlr.csail.mit.edu/proceedings/papers/v28/goodfellow13.pdf, top of second page, ...
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 $p = \infty$ ?
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 ...
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
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, ...
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 ...
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 ...
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 $A=B^{-1}B^*$?
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 & ...
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
46 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
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
16 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) ...
0
votes
2answers
36 views

Determinant of symmetric matrix $(A-\lambda I)$

If we have a matrix $(A-\lambda I)$ which is: $\left( \begin{array}{ccc} 1-\lambda & -1 & 2 \\ -1 & 1-\lambda & 2 \\ 2 & 2 & 2-\lambda \\ \end{array} \right) $ Then it's ...
-1
votes
1answer
50 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 ...
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
0answers
22 views

${A^{ - 1}}$ is similar to $A^*$.why there is a nonsingular $B$ such that $A=B^{-1}B^*$? . [on hold]

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

Dimension of Vector Spaces

Can anybody help me finding out the dimension of the vector spaces: A: A is $n\times n$ real upper triangular matrices. A: A is $n\times n$ real symmetric matrices. A: A is $m\times n$ real ...
1
vote
2answers
39 views

Geometrical Interpretation of Matrix Multiplication

I am stuck up with this question from my Linear Algebra Assignment which states to explain geometrically why $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix}-1 & 0 \\ 0 ...
2
votes
2answers
45 views

Proving that the matrix is positive definite

I have looked at similar questions under 'Questions that may already have your answer" and unless I have missed it, I cannot find a similar question. I am trying to answer the following: Let $A = ...
0
votes
3answers
38 views

Some questions about notation in “$[T]_\alpha^\beta$”

I just have a few questions about the general meaning of the notation "$[T]_\alpha^\beta$". I would really appreciate if someone would dumb it WAY down to the most basic level (no assumptions, no ...
2
votes
1answer
25 views

Confusion with the notation $L_A$

My linear algebra class went from 0-100 real quick. I've attended every single lecture (so I know I haven't missed out on anything); however, very recently he has been using the notation $L_A$ for a ...
1
vote
1answer
24 views

Cokernel as direct sum of cyclic groups

I am asked to reduce the matrix $ \left( \begin{array}{ccc} 3 & 1 & -4 \\ 2 & -3 & 1 \\ -4 & 6 & -2 \end{array} \right)$ to diagonal form over $\mathbb{Z}$ and then write the ...
-1
votes
0answers
20 views

$A$ is normal. Why there is polynomial $p(t)$ of degree at most $n-1$ such that $A^*=p(A)$? . [closed]

Let $A \in {M_n}$ and $A$ is normal. Why there is polynomial $p(t)$ of degree at most $n-1$ such that $A^*=p(A)$?
-2
votes
1answer
22 views

There is a polynomial $p(t)$ such that $A^*=p(A)$. Why $A$ is normal [closed]

Let $A \in {M_n}$ and there is a polynomial $p(t)$ such that $A^*=p(A)$. Why $A$ is normal?
1
vote
3answers
53 views

For which values does the Matrix system have a unique solution, infinitely many solutions and no solution?

Given the system: $$\begin{align} & x+3y-3z=4 \\ & y+2z=a \\ & 2x+5y+(a^2-9)z=9 \end{align}$$ For which values of a (if any) does the system have a unique solution, infinitely many ...
0
votes
4answers
41 views

Why if the columns of a matrix are not linearly independent the matrix is not invertible?

Why if the columns of a matrix are not linearly independent the matrix is not invertible? I have watched this video about eigenvalues and eigenvectors by Sal from Khan Academy, where he says that for ...
0
votes
2answers
33 views

Show that V is a vector space over the set of real numbers when V is the set of all real 3x3 matrices

Wondering how one would go on about this. V is the set of all real 3 × 3 matrices. How can it be shown that V is a vector space over the set of real numbers and what would be the dimension of and ...
0
votes
1answer
15 views

How do you find the matrix relative to a basis?

I'm having trouble knowing where to start. I've been given the problem: Let $\ B = \{1, x, sin(x), cos(x)\}$ be a basis for a subspace $\ W$ of the space of continuous functions, and let $\ Dx $ be ...
0
votes
0answers
23 views

Show: If $v \in E^{\perp}$ then it can be written as $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$

(i) Assume that $B = \{w_1,\dots,w_k\}$ is an orthogonal basis for $E$. Let $v \in E^{\perp}$ such that $v\neq O_{V}$. Prove that $v=w+c_1w_1+c_2w_2+\dots +c_kw_k$ for some nonzero $w\notin E$ and ...
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]; ...
4
votes
3answers
105 views

Skew-symmetric matrix subspace dimension and basis

If $M$ is the vector space of $2\times 2$ real matrices, then I can show that $$ \{A \in M \mid A^\mathrm{T}=-A \} $$ is a subspace of $M$, since $$ \left[ \begin{array}{cc} x & z \\ -z & ...
1
vote
1answer
32 views

Find the Jordan form of a 4 x 4 matrix

Find the Jordan Form of $$ A=\left[\begin{array}{cccc} 0 & -16 & 0 & 0\\ 1 & 8 & 0 & 0\\ 0 & 0 & 0 & -6\\ 0 & 0 & 1 &5 \end{array}\right] $$ ...
0
votes
2answers
16 views

Orthogonal Projection in subspace

Consider the vector space $\mathbb{R}^n$ with usual inner product $<.,.>$. Take $Y\in \mathbb{R}^n$ and $X \in \mathbb{R}^n$ such that $Y=[y_1,y_2,..y_n]^t$ and $X=[1,1,....1]^t$ ...
6
votes
2answers
106 views

Finding minimum from matrix

Consider following $3\times 3$ matrix. $\begin{pmatrix}3&6&9\\ 2& 4 &8\\ 1 &5& 7 \end{pmatrix}$ I need to find combination of three numbers where each number ...
0
votes
1answer
19 views

Find the coordinate matrix of a polynomial with respect to a non-standard basis

I'm stuck on this question here: Find the coordinate matrix of $2-4x-3x^2$ with respect to $B = {2, x^2-1, 1-2x-x^2}$ I did the following: $a(2) + b(x^2 - 1) + c(1-2x-x^2) = 2-4x-3x^2$ But now I'm ...
-4
votes
1answer
34 views

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

Let $A,B \in {M_n}$ and they have the same singular values.Why $A$ and $B$ are unitary equivalent?(by SVD)
2
votes
2answers
39 views

What is the difference between $n$-tuples, $m \times 1$ and $1 \times n$ matrices?

Isn't the tuple different structure from $m \times 1$ or $1 \times n$ matrix? Why can you mix them?
-2
votes
0answers
32 views

Rotation quaternions and rotation matrix for electron spin [on hold]

Is it possible to construct a rotation quaternion and rotation matrix for the approximation of an electron spin?
-3
votes
0answers
20 views

If $AA^*$ and $BB^*$ are unitary equivalent. Is it true that $A$ and $B$ are unitary equivalent? [closed]

Let $AA^*$ and $BB^*$ are unitary equivalent. Is it true that $A$ and $B$ are unitary equivalent?
0
votes
1answer
43 views

Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$.

Let , $A_{6\times 6}$ diagonal matrix with characteristic polynomial $x(x+1)^2(x-1)^3$. Find the dimension of the set $S=\{B_{6\times 6}: AB=BA\}$. From characteristic polynomial of $A$ , first ...
0
votes
1answer
26 views

What is the Lie group that leaves this matrix invariant?

What is the group that leaves \begin{equation} Y = \bigoplus_{j=1}^{k} \alpha_j \mathbb{I}_{2n_j\times 2n_j} \end{equation} invariant under congruence ($Y = XYX^T$) where $\alpha_j \in \mathbb{R}$ ...
0
votes
1answer
32 views

Solve matrix equation $AXB+CX=D$

How to solve matrix equation $AXB+CX=D$ for $X$? If it is not solvable, are there any numerical methods to do it?
1
vote
2answers
24 views

find the smallest interval in which the eigen value of the matrix lie

$$ \begin{bmatrix} 3 & 2 & 2 \\ 2 & 5 & 2 \\ 2 & 2 & 3 \\ \end{bmatrix} $$ I was practicing questions on Matrices & Determinants ...