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), ...

learn more… | top users | synonyms (2)

2
votes
2answers
47 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
39 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
26 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
55 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
18 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
20 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
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)
2
votes
2answers
40 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
33 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 ...
0
votes
2answers
46 views

if $rank{(A - \lambda I)^k} = rank{(B - \lambda I)^k}$ then $A$ is similar $B$

Let $A,B \in M_n(\mathbb{R}).$ Suppose for all $\lambda \in \sigma (A)$ and for all $k \geq 0,$ we have $\mathrm{rank}(A - \lambda I)^k = \mathrm{rank}(B - \lambda I)^k.$ Then why are $A$ and $B $ ...
0
votes
1answer
58 views

If each eigenvalueof $A$ is either $+1$ or $-1$ $ \Rightarrow$ $A$ is similar to ${A^{ - 1}}$

Let $A \in {M_n}$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ is similar to ${A^{ - 1}}$?
0
votes
1answer
27 views

Eigenvalues of an 2x2 matrix [closed]

How do i calculate the EigenValues of an Hessian Matrix which is 2x2.? And what is EigenValues. It is used in imagesProc. when i have to find goodFeatures is found when both eigenvalues are high.. but ...
1
vote
1answer
36 views

Jordan form of different matrices

Suppose you have a 4x4 matrix with the characteristic polynomial equal to the minimal polynomial $m_F(x)=C_F(x)=(x-3)^2(x+2)^2$. Find the Jordan form. Is this the correct solution? $$ ...
1
vote
1answer
16 views

A question about non-linear least square method…

I am trying to fit a set of points into a sine function, using nonlinear least square method. The final step to obtain the derivative of its parameters is given by the equation (8) of: ...
0
votes
0answers
36 views

How to calculate a person's Latitudinal and Longitudinal location based off of Sun and time

INTRO I remembered hearing about it being possible to calculate a person's position or the position a picture was taken, based on time of day and the position of the Sun, position meaning latitudinal ...
2
votes
0answers
14 views

Cartan matrices: motivation and intuitive examples?

could anyone provide me with a sketch of the motivation that gave rise to Cartan matrices in abstract (homological) algebra, Lie algebrae and so on? Which was the trigger or the need for them? It ...
0
votes
0answers
17 views

Can matrix generated by ith power of adjacency matrix, have -ve value?

I read that - The uv-entry of the k-th power $$A^k$$ counts the number of walks of length k from the vertex u to the vertex v. I wanted to know if such a matrix can have negative values, and ...
-3
votes
0answers
31 views

$A=UMW^*$ and $B=VMO^*$.Is this true that $A$ and $B$ are unitarily equivalent? [on hold]

Let $U,W,V,O$ are unitary matrises and $A=UMW^*$ and $B=VMO^*$.Is this true that $A$ and $B$ are unitarily equivalent?(all matrises n-by-n)
-1
votes
1answer
45 views

$A,B $ are both normal and $A$ and $B$ commute.Why $AB$ is normal? [closed]

Let $A,B \in {M_n}$ are both normal and $A$ and $B$ commute.Why $AB$ is normal?
0
votes
1answer
25 views

Complete misunderstanding of Lie groups and representations

Consider a particular representation of $\operatorname{SO}(2,\mathbb{R})$: \begin{equation} \begin{pmatrix} \operatorname{cos}(\theta) & \operatorname{sin}(\theta) \\ ...
0
votes
1answer
37 views

How can we find if a matrix is full column rank

If $A$ is an $n*k$ matrix with complicated form of elements. How can I show this matrix is full column rank? By complicated form I mean there is no known form for the elements of $A$.
1
vote
1answer
42 views

Find Jordan Form of αA (α is a scalar, A a matrix)

In my linear algebra course I have a problem which goes as follows: Suppose A is an nxn matrix over field (R) And J(A) is the jordan form of A. Given α belongs to field R, what is the jordan form of ...
0
votes
0answers
19 views

Signature Defect of a Matrix

Let $A,D$ be symmetric real matrices and let $B,C$ be real matrices such that $B$ and $C$ have the same number of rows, $A$ has the same number of columns as $B$, and $C$ has the same number of ...
0
votes
1answer
23 views

Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)? .

Let $A \in {M_n}$.Is this true that $A$ and $A^*$ are unitary equivalent(or equivalent)?
-2
votes
0answers
27 views

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

Let $A,B \in {M_n}$ and they are same singular values.Why $A$ and $B$ are unitary equivalent?
1
vote
0answers
24 views

Formula for powering a matrix not working for all matrices

I'm currently learning about matrices and was asked to show that this formula works for powers of $M$. $$M^n = nM-(n-1)I$$ Where $M$ is the matrix (show below), $n$ is the exponent an $I$ is the ...
0
votes
2answers
21 views

proof a theorem in linear algebra

prove that if λ1 and λ2 are two distinct eigenvalues of a matrix A and λ1 , λ2 are corresponding eigenvectors, respectively, then α1 and α2 are linearly independent please help... thank you...
0
votes
1answer
40 views

If ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = \sum\limits_{i = 1}^n {{\sigma _i}^2} \Rightarrow$A is normal matrix

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$ and suppose ${\sum\limits_{i = 1}^n {\left| {{\lambda _i}} \right|} ^2} = ...
-1
votes
0answers
12 views

simple question in $QR$ [closed]

Let $A \in {M_n}$ and let $A=QR$ be a $QR$ factorization. suppose $A$ is normal matrix why $RQ$ is normal?
1
vote
2answers
48 views

Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?

Let $A \in {M_n}$ have eigenvalues ${\lambda _1}.....{\lambda _n}$ and singular values ${\sigma _1}.....{\sigma _n}$. Why does $tr({A^*}A) = \sum\limits_{i = 1}^n {{\sigma _i}^2} $?
1
vote
2answers
89 views

How is the vector space of abstract “tuples” isomorphic to vector space of $n \times 1$ or $1 \times n $ matrices?

I read that the vector space of abstract "tuples" is isomorphic to vector space of $n \times 1$ or $1 \times n $ matrices. Where can I find a good explanation of this or can someone explain it here?
2
votes
2answers
58 views

If $\dim_F F[A] <n$, then $A$ is not cyclic. [closed]

Suppose that $A \in M_n(F)$ and the minimal polynomial of $A$ is irreducible. S0 $F[A]$ is a field extension of $F$. I have to show: 1) If $\dim_F F[A] <n$, then $A$ is not cyclic. 2) If $\dim_F ...
1
vote
1answer
20 views

finding matrix represention for linear transformation for field extension

need some clarification. given an extension field K over F with F-linear transformation, for $\alpha \in K$, $f_\alpha(k) = \alpha \cdot k$ i.e. multiplication on the left. I need to find the ...
1
vote
0answers
12 views

Similarity of orthogonal matrices

Prove that for any $M$ in $SO(3)$, there is a matrix $P$ in $SO(3)$ and a real $\alpha$ such that $$PMP^{-1} = \left[ \begin{matrix} \cos\alpha & \sin\alpha & 0 \\ ...