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
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1answer
55 views

Finding jordan form and rational form of a $5x5$ matrix.

Consider $$A=\begin{bmatrix} 0 & 0 & 2 & 0 &0 \\ 0 & 0& 0 & 6& 0\\ 0&0&0&0&12 \\ 0&0&0&0&0\\ 0&0&0&0&0 ...
0
votes
1answer
42 views

Rank of the given matrix [closed]

Let $A=(A_{ij})$ be a matrix of order $n$, where $A_{ij}= i+j$. Find the rank of $A$.
0
votes
0answers
56 views

Is it possible to resolve equations of two vectors

I have a objective function as following $$F=\int |\alpha^TG(x)-w^TJ(x)|^2 H(x)\,dx+\lambda_1 \alpha^2+\lambda_2 w^2$$ where $\alpha^T$ is transpose of vector $\alpha= \begin{bmatrix} ...
1
vote
1answer
191 views

Upper bound of Frobenius norm of product of matrices.

I'm trying to prove that $||AB||_F\leq||A||_2||B||_F$. As far as I know it isn't a hard problem but I was stuck. Any ideas?
0
votes
2answers
23 views

Can i have a column filled with zeroes when computer $T$?

Let $T: P2 \rightarrow P2$ be the liner transformation defined by $T(p(x)) = xp'(x)$. Find the matrix $A= [T]$ of $T$ relative to the standard basis of $P2$ : B=$(1,x,x^2) $ My answer basically was: ...
2
votes
1answer
107 views

matrices with determinant equals to one

we already know what does it mean the determiant of a matrix is null, it's not invertible ! but what about matrices with determinant equals to $1$ ?! I know that the determinant of matrix is the ...
4
votes
1answer
2k views

Proof of Vandermonde Matrix Inverse Formula

I'm working through Exercise 40 from section 1.2.3 of Knuth's The Art of Computer Programming volume 1, but am finding myself unable to produce a rigorous proof, and the one here is suspect and not ...
0
votes
2answers
67 views

How to prove that A is diagonalizable?

So I was given this question in my exam, and it is by no mean a homework. Let A = \begin{pmatrix} 5 & 1 & 0 \\ 1 & 5 & 0 \\ 0 & 0 & 6 \end{pmatrix} (a) Find the ...
3
votes
3answers
142 views

Determine the coefficient of polynomial det(I + xA)

Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues. How do I determine the coefficient of the term $x^2$ of the polynomial given by $q(x) = \det(I_n + xA)$
2
votes
0answers
30 views

Trick to rewrite operator in terms of another?

In the book Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems by Bill Sutherland, I would like to understand the trick done in (4), see the excerpt from p 29 shown below I ...
13
votes
1answer
533 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
7
votes
2answers
591 views

How many 1's can a regular 0,1-matrix contain?

A matrix of order $n$ has all of its entries in $\{0,1\}$. What is the maximum number of $1$ in the matrix for which the matrix is non singular.
2
votes
2answers
70 views

showing that 2 matrices are not similar

There are two $3\times 3$ matrices: $$ A = \begin{bmatrix} 2 &-1 &-1\\ 0& 1 &1\\ 0 &0 &2 \end{bmatrix} $$ $$ B = \begin{bmatrix} 2 &-1 &1\\ 0& 1 &1\\ ...
0
votes
0answers
11 views

State space form

I want to write the following equations on state space form, but I am unsure how it looks. $u_t=u_t^*+u_t^*+\epsilon_t^1$ $u_{t+1}^*=u_{t}^*+\epsilon_t^2$ $u_{t+1}^c=\lambda_1 u_{t}^c+\lambda_2 ...
0
votes
0answers
48 views

Detrrminant of a matrix

I need to prove that $\det A_n \ne 0$, if $a_i \ne a_j$ and $a_i \ne 0$, where $A_n$ is a matrix with the first row $a_1 +a_2, a_2+a_3, \ldots , a_{n-1}+a_n , a_n$. Second row consists of all ...
1
vote
1answer
21 views

Counterclockwise rotation matrix

If I take the basis $(\vec{e_x},\vec{e_y})$ and make a rotation counterclockwise of angle $\theta$, I end up with two new vectors $(\vec{u},\vec{v})$ such that : $\vec{u} = \cos\theta \vec{e_x} + ...
2
votes
2answers
125 views

For what values of k is this singular matrix diagonalizable?

So the matrix is the following: \begin{bmatrix} 1 &1 &k \\ 1&1 &k \\ 1&1 &k \end{bmatrix} I've found the eigan values which are $0$ with an algebraic multiplicity of $2$ ...
2
votes
1answer
44 views

Vandermonde determinant and linearly independent (corrected version)

This is a corrected version. Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 ...
0
votes
2answers
93 views

Determining if vector space holds

Let A be a particular vector in $\Bbb R$2x2. Determine whether the following is a subspace of $\Bbb R$2x2: S = {B ∈ $\Bbb R$2x2 | AB + B = O} I have two ideas on how to approach this for scalar ...
0
votes
1answer
88 views

a multiple choice question related to trace of a matrix.

let P and Q are two invertible matrices . and PQ= -QP . then which of the following is true a) trace(P)=trace(Q)=0 c)trace(P) is not equal to trace(Q) c) none of the above. i can show that ...
1
vote
1answer
63 views

What does do the bases of the eigenvectors of a 2x2 matrix say about the nature of the matrix?

Hi this is question I'm having trouble for while preparing for my test. I have matrix: R = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} I found it to have eigenvalues of -1 and 1 for 1 the ...
0
votes
0answers
42 views

Interpreting a diagonalized matrix

I'm doing a practice question before a test: The stress in a solid at a point P can be described by a matrix T called the stress matrix (or stress tensor). If n is a normal vector to a plane cutting ...
0
votes
1answer
37 views

Absorbing Markov chain when less transient states than absorbing states

I have a probability matrix: 1 2 3 1 0.5 0.3 0.2 2 0 1 0 3 0 0 1 I understand that: $$ Q = \left(\begin{array}{c} 0.5 \end{array} ...
1
vote
2answers
101 views

Matrix diagonalisable in R, but not in C.

I know is quite easy to find a matrix $A\in\mathbb{R}^{2,2}$ that is diagonalisable if the base field is $\mathbb{C}$, but not diagonalisable if the base field is $\mathbb{R}$. The easiest example can ...
3
votes
0answers
75 views

Largest Singular Value / Singular Value

I was wondering, what if the eigenvalues of a matrix A are all negative. So does that simply mean there is no singular value for this particular matrix?, hence I can't calculate the conditional number ...
2
votes
2answers
67 views

Notation for replacing a matrix column with a vector

Let $A$ be an $n\times n$ matrix. Let $v$ be an $n\times 1$ matrix. Is there a notation to signify replacing the $j$-th column of $A$ with $v$? If not, what is the accepted way to denote this?
1
vote
3answers
88 views

When should matrices have units of measurement?

As a mathematician I think of matrices as $\mathbb{F}^{m\times n}$, where $\mathbb{F}$ is a field and usually $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$. Units are not necessary. However, ...
0
votes
1answer
24 views

Finding eigenvalues/vectors of a matrix and proving it is not diagonalisable.

I have got the following matrix. $$\begin{pmatrix} -7 &4 \\ -9 &5 \end{pmatrix}$$ I need to find the eigenvalues, eigenvectors and $\textbf{prove}$ that it is not diagonalisable. I have ...
1
vote
2answers
118 views

In how many ways can a $5 \times 5$ matrix be formed such that sum of row elements and column elements are $4$ and entries are $0$ or $1$?

Let we have a $5 \times 5$ matrix and the elements can be either $0$ or $1$ and the sum of elements of each row and column is $4$ then in how many ways can the matrix be formed ? I tried doing it in ...
2
votes
4answers
167 views

Proving that $\operatorname{rank}(AB)$ is smaller or equal to $\operatorname{rank}(B)$ [duplicate]

I am struggling with proving the theorem that if $A$ and $B$ are $n\times n$ matrices, then: $$\operatorname{rank}(AB)\leq \operatorname{rank}(B)$$ Could anyone suggest me a hint? Any help is ...
3
votes
2answers
66 views

Finding a nullspace of a matrix.

I am given the following matrix $A$ and I need to find a nullspace of this matrix. $$A = \begin{pmatrix} 2&1&4&-1 \\ 1&1&1&1 \\ 1&0&3&-2 \\ ...
0
votes
1answer
98 views

Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ ...
2
votes
3answers
155 views

Using detA and detB to calculate the determinant of matrix C

If we have C=($A^t$)$^2$BA$^3$B$^-$$^1$A$^-$$^3$ and detA=-2 and detB doesnt equal 0, how do we calculate det C? I know that the transpose of a matrix does not affect the determinant. Does this mean ...
18
votes
11answers
527 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? [duplicate]

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
1
vote
0answers
64 views

Matrix transponse in tensor notation

In this paper http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf at the end of chapeter 2 the author says that in index notation a matrix is written as $A^\mu_{\;\;\nu}$ and its ...
0
votes
3answers
130 views

Find XY given matrix YX where X is a row matrix and Y is a column matrix

I've been given matrix $YX$ as below. I want to find $XY$ from it. I know that X is a row matrix and Y is a column matrix. $X$ has 2 entries and $Y$ has 2 entries. However I don't know the values of ...
15
votes
3answers
3k views

Recovering the two $SU(2)$ matrices from$ SO(4)$ matrix

Since there is a $2$-$1$ homomorphism from $SU(2)\times SU(2)$ to $SO(4)$ there should be a way to recover the two $SU(2)$ matrices given an $SO(4)$ matrix. I believe I could set this up as a ...
0
votes
2answers
43 views

How to show that $T$ is linearly independent?

This question came in my exam, and is by no mean a homework. Let $\{v_1,v_2\}$ be a linearly independent subset of a vector space $V$ and let $w$ belong to $V$ but not to ...
2
votes
1answer
2k views

Rotation Matrices - Rotating a point on a graph

I'm trying to understand how rotation matrices work in Linear Algebra... I don't think I'm visualizing it correctly though... I'd like to rotate a point (-2, 1) around a graph... the point (-2, 1) ...
2
votes
1answer
83 views

Vandermonde determinant and linearly independent

Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 & a_2^{2} & a_3^{2} & ...
0
votes
0answers
17 views

find mean of matrices $A_i, A_j$ given $d_{A_{ji}}=\ln{\left|\left| A_{ji} \right|\right| \left|\left| A_{ji}^{-1} \right|\right|}$

Given a finite set $\mathbb{A}$ of $k$ like-shaped, square, non-singular matrices $A_i\in\mathbb{R}^{n\times n}$, let's define $A_{ji}=A_j A_i^{-1}$, then the distance of the two matrices $A_i, A_j$ ...
2
votes
1answer
362 views

Diagonalize tri-diagonal symmetric matrix

How to diagonalize the following matrix? \begin{pmatrix} 2 & -1 & 0 & 0 & 0 & \cdots \\ -1 & 2 & -1 & 0 & 0 & \cdots \\ 0 & -1 & 2 & -1 & 0 ...
-1
votes
1answer
81 views

Finding a pair of orthogonal vectors in $R^4$

Find a pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3). What i have tried so far:
0
votes
0answers
92 views

$\det(AB) = 0$ when $A$ has more rows than $B$

Someone posted here that when $A$ has more rows than $B$, $\det(AB)=0$. How?
0
votes
1answer
32 views

Let T be a one-to-one linear transformation from $R^m$ to $R^n$ and B={$e_1$,$e_2$,…,$e_m$} a basis for $R^m$.

Prove that the set {T($e_1$),T($e_2$),...,T($e_m$)} is an independent set. Let T : $R^n$ → $R^m$ be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in ...
1
vote
0answers
24 views

Why are the eigenvalues of $I_R + \beta' \alpha$ bounded by one?

Trying to understand the Granger-Johansen Representation Thm (see p. 7 here; we are assuming Condition 5). We have $(p \times r)$ matrices $\alpha, \beta$. We know that $|eig(a \beta')| \leq 1$ and ...
1
vote
2answers
41 views

Least Square method, find vector x that minimises $ ||Ax-b||_2^2$

Given Matrix A = | 1 0 1 | | 1 1 2 | | 0 -1 -1| and b = $[1\ \ 4\ -2]^T$ find x such that $||Ax - b||_2^2$ is minimised. I know I have to do something along the line $A^TAx = A^Tb$ got the ...
2
votes
1answer
55 views

Which matrices are covariances matrices?

Let $V$ be a matrix. What conditions should we require so that we can find a random vector $X = (X_1, \dots, X_n)$ so that $V = Var(X)$? Of course necessary conditions are: All the elements on ...
-1
votes
1answer
43 views

Why do the columns of the inverse of a matrix (defined as a linear operator) form an orthogonal basis in an inner product space?

Let V be a vector space over C and W be an inner product space over C with inner product <., .> and T:V --> W be a linear transformation. Find an orthogonal basis for V = R^3 with the inner product ...
6
votes
1answer
196 views

Solving a linear equation given the solution of another

Suppose I have a matrix $S$ having a one-dimensional nullspace $\{ e \}$ such that $S + ee^\top$ is a positive definite symmetric matrix. Now let $b \in Range(S)$ and suppose I solve the equation $(S ...