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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1answer
18 views

How to find worst case in chain matrix multiplication

The question we got was Determine a worst-case parenthesization of the matrix-chain product whose sequence of dimensions is (5, 2, 3, 10, 4, 6, 7, 8). what i dont understand is how do we determine ...
0
votes
1answer
13 views

How can it be shown that $\vec{w} + \vec{v}$ is either an eigenvector of a symmetric matrix or equal to the zero-vector?

How can it be shown that $\vec{w} + \vec{v}$ is either an eigenvector of H or equal to the zero-vector? I'm not sure how to approach this. Here are the details given: I is a 3x3 identity matrix. P ...
3
votes
0answers
41 views

If I know $AB$ (that is symetric), how can I calculate $BA$. $A∈\mathscr{M}_{3×2}(\mathbb{R})$ and $B∈\mathscr{M}_{2x3}(\mathbb{R})$

Let $A∈\mathscr{M}_{3×2}(\mathbb{R})$ and $B∈\mathscr{M}_{2x3}(\mathbb{R})$ be matrices satisfying $AB =\begin{bmatrix} 8 &2 &−2\\ 2 &5 &4\\ −2 &4& 5 \end{bmatrix}$. Calculate ...
0
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0answers
12 views

Lower bound on block-matrix

Let $\mathbb{R}^{n \times n} \ni C(x) = C(x)^\top \succ 0$ for all $x \in \mathbb{R}^n$. Let $A \in \mathbb{R}^{m \times n}$, where $\text{rank}(A) = m$, $m \leq n$. Further, let $\mathbb{R}^{n \times ...
0
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0answers
18 views

derivative of a scalar wrt matrix

Let $y = \|A^T\mathbf{x} + \mathbf{b}\|_2^2$ where A is a matrix of size $d \times D$, $\mathbf{x}$ and $\mathbf{b}$ are $d\times 1$ vectors. What is the derivative of y wrt A? Is it ...
-4
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0answers
18 views

jordan canonical form… [on hold]

I don't understand how to apply jordan canonical form of a matrix. Please Guide me with some examples or some reference regarding it. Thanks...
2
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0answers
8 views

Converting a max-min problem to a max problem with a constraint

The objective is to find the greatest lower bound of the variable $\mu$. The lower bound is resulting from the positive-semidefinite (PSD) constraint $$\tilde{\mathbf{T}}:=\left( \begin{array}{ccc} ...
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3answers
32 views

Cramer's Rule Proof Question

I have read the following proof on Wikipedia How does $X_1$ columns are $A^{-1}b,A^{-1}v_2,...,A^{-1}v_{n'}$ are they to columns augmented? or are they matrix multiplication ?
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0answers
25 views

A special argument to derive the derivative of the determinant

In this short note, in the section "a better result" the author says: [...] if $\Phi(t)$ is the identity [...] then $$ \frac{d}{d t} \operatorname{det} \Phi(t) = \operatorname{tr} \dot{\Phi}(t) ...
2
votes
2answers
33 views

Let $A$ is nonsingular and each eigenvalue of $A$ is either $+1$ or $-1$.Why $A$ 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 ${A^{ - 1}}$?
5
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0answers
26 views

Counterexamples to the Matrix norm AM-GM inequality?

I am new here and this my first question, I hope I am being as clear as possible and apologize in advance for any misunderstandings. I am researching the Arithmetic-Geometric Mean (AM-GM) inequality ...
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vote
3answers
33 views

transforming $(A,B,C)$ to $(0, 0, 1)$ by rotations

I'm trying to reflect the "world" through a specified plane $p:Ax+By+Cz=0$. I know how to reflect the "world" through the $xy$-plane, so I want to rotate $p$ in the $3$ axes ($x,y,z$-axes) so it will ...
0
votes
0answers
12 views

Matrix Properties Reference

A lot of proofs I come across (working on stability of numerical methods) apply some property of a particular type of matrix. This includes, for example, the fact that the $L_{2}$ norm of a normal ...
0
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1answer
24 views

Matrix Change of Basis

guys. I'm not entirely sure how I'm not getting the right answer for this question. I'll try to explain what I've tried so far. I need to computer MB1->B2 and MB2->B1 B1 = {(0,0,1),(1,0,0),(0,1,0)} ...
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0answers
7 views

Powers of a defective matrix

For non-defective matrices you can calculate arbitrary powers by diagonalizing it. Is there an easy way to calculate the powers of a defective matrix?
5
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1answer
41 views

Compare determinants of matrices with different dimensions

Reading about matrices and determinants I am wondering about the following concept: How valid is to compare the determinants of matrices with different dimensions? e.g. compare a determinant $D1$ ...
2
votes
1answer
36 views

Central extension of the Discrete Heisenberg group $H_3(\Bbb Z)$

I want to use the Discrete Heisenberg group $(H_3(\Bbb Z),\times)$ as an example for a presentation on central extensions. $H_3(\Bbb Z) = \begin{bmatrix}1&x&z\\0&1&y\\0&0&1 ...
4
votes
3answers
52 views

Give an example of a singular matrix in $M_{3×3}(Q)$ the entries of which are distinct prime positive integers, or show that no such matrix can exist.

I know that the matrix exist because the entries are primes but I don´t know how to explain, i need some help. Give an example of a singular matrix in $M_{3×3}(Q)$ the entries of which are distinct ...
0
votes
0answers
35 views

Is $(A-A^{-1})$ skew-symmetric?

If $A$ is orthogonal, $(A-A^{-1})^T=A-A^T\neq -(A-A^{-1})=A^{-1}-A$ If $A$ is involutory, do we have an exception? In that case $(A-A^{-1})=0$, which seems trivial.
2
votes
1answer
32 views

Similar matrices NOT over the complex numbers [duplicate]

We say that two matrices $A,B$ with complex entries are similar if and only if there exists an invertible complex matrix $P$ so that $A = P^{-1} B P$. Does $P$ always have to be a complex matrix? ...
1
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0answers
19 views

Is the action of the matrix UU^(t) always a projection? What can I say about I - UU^t?

The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal ...
0
votes
1answer
38 views

Product of upper-triangular matrices.

** I´m trying to solve this problem, but I don´t now how to start, I think could be by induction but I´m not sure. ** Let $n$ be a positive integer and let $F$ be a field. Let $A_1, . . . , ...
0
votes
1answer
15 views

Does $AS=SB\iff f_A(\lambda)=f_B(\lambda)$?

Showing the converse is straightforward: $$B=S^{-1}AS\Rightarrow f_B(\lambda)=\det(B-\lambda I_n)=\det(S^{-1}AS-\lambda I_n)=\det(S^{-1}(A-\lambda I_n)S)\\=(\det S)^{-1}\det (A-\lambda I_n)\det ...
1
vote
0answers
9 views

A proof for a theorem related to rank and matrix product. [duplicate]

For all matrix $\mathbf{M} \in \mathbb{R}^{m,n}$ and $\mathbf{N} \in \mathbb{R}^{n,p}$, the inequality $\operatorname{rank}\mathbf{M} + \operatorname{rank}\mathbf{N} - n \leq ...
1
vote
3answers
32 views

Is any linear transformation with $\text{ker }(T)=\left\{\vec{0}\right\}$ an isomorphism?

I'm thinking no; for instance, $\exists \left\{\left.T:V\rightarrow W\right| \text{Im }(T)\neq W\right\}$. This seems counterintuitive, though. If such a $T$ with maximal rank exists, What would ...
0
votes
2answers
18 views

What is the maximum value of $\text{dim ker }A$, where $A$ is $n\times m$?

True or false: "If $A$ is an $n\times m$ matrix, then $\text{dim ker }A\leq n$" My gut intuitively tells me "no"$\,\Rightarrow$ if $m>n$, $\text{dim ker }A\leq m$. I can't think of a simple, ...
3
votes
3answers
55 views

Wouldn't each addition take time $O(n)$?

I am going over the asymptotic runtime of regular matrix multiplication. Here is a lecture slide I am referencing(too much to type out, shown below), from Algorithms Everything makes sense up ...
1
vote
1answer
18 views

Let F be a field and let $A,B ∈M_{n×n}(F)$ be a commuting pair of matrices, where B is nonsingular. Is $(A,B^{−1})$ necessarily a commuting pair?

I´m trying to solve this problem, but I can´t, I don´t know how to start. Let F be a field and let $A,B ∈M_{n×n}(F)$ be a commuting pair of matrices, where B is nonsingular. Is $(A,B^{−1})$ ...
2
votes
1answer
17 views

According to Buckingham Theorem the rank of $A$ should be $2$

A physical system is described by a law of the form $f(E,P,A)=0$, where $E,P,A$ represent, respectively, energy, pressure and surface area. Find an equivalent physical law that relates suitable ...
0
votes
0answers
12 views

decomposing multiplication of two matrices to the sum of rank-1 matrices

Suppose we have two matrices: $D_{n \times k}$ and $X_{k \times p}$ I need to understand how do we decompose the multiplication DX to the sum of $k$ (am I correct about $k$?) rank_$1$ matrices. ...
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votes
1answer
48 views

Determinant Calculation Issue

Solved..found my mistakes.Thanks David for pointing out the first one to made me realize the other problem in C. I was asked to calculate the determinant for the following matrix: \begin{matrix} ...
0
votes
1answer
26 views

Find $a$ and $b$ in a 4 equation system

$a, b \in\mathbb{R}$. I have four equations: $$x+3y-2z+t=-3$$ $$3x+11y+az+5t=2$$ $$3x+12y-6z+6t=b$$ $$4x+15y-8z+8t=-5$$ I have to find out the values of $a$ and $b$ where the system is solvable (has ...
1
vote
1answer
34 views

Show that a nonzero $2 \times 2$ matrix $A$ such that $A^2 = 0$ is similar to $\begin{pmatrix}0&1\\0&0\end{pmatrix}$

Let $A$ be a $2 \times 2$ non-zero matrix such that $${A}^{2}=0.$$ How do I find an invertible matrix P such that $${P}^{-1}AP=\begin{bmatrix}0&1\\0&0\end{bmatrix} ?$$ Anyone? Please provide ...
2
votes
1answer
59 views

The only eigenvalue of $A \in {M_n}$ is $\lambda = 1$. Why is $A$ similar to $A^k$?

Suppose that the only eigenvalue of $A \in {M_n}$ is $\lambda = 1$. Why is $A$ similar to $A^k$ for each $k=1,2,3,\dots$?
0
votes
0answers
13 views

Differntiating matrix functions $f : \mathbb R^{n\times m} \to \mathbb R^{p\times q}$

How would you differentiate matrix functions $f : \mathbb R^{n\times m} \to \mathbb R^{p\times q}$ like for example $f(X) = X^T \cdot X$? There are no directional derivatives in the usual sense, and ...
0
votes
1answer
22 views

Calculating the adjoint

I am having some trouble understanding the idea of cofactors and adjoints of matrices. From my understanding the adjoint of a matrix is the transpose of the matrix of cofactors? $A=\begin{bmatrix} 1 ...
0
votes
0answers
24 views

Compute det(A) given a function A

Suppose A is a 3×3 matrix and A = 1/3 $u_1\cdot uT_1$ + 1/4 $u_2\cdot uT_2$ + 2/5 $u_3\cdot uT_3$ with $uT_1 = (0, 1, −1)$ $uT_2 = (1, 2, 2)$ $u_3 = (−2,1/2,1/2)$ Compute det(A). I know ...
2
votes
1answer
50 views

Let $\text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}$. Why are $A$ and $B$ similar?

Let $A$ and $B \in M_n$ be two matrices such that $$\forall k=1,2,\dots,n,\ \forall \lambda\ \text{eigenvalue of $A$},\ \text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}.$$ Why are $A$ ...
-1
votes
2answers
11 views

$A$ is similar to $cA$ for some complex scaler with $\left| c \right| \ne 1$.why dose all eigenvalue of matrix $A$ are zero?

Let$A \in {M_n}$ and $A$ is similar to $cA$ for some complex scaler with $\left| c \right| \ne 1$.why dose all eigenvalue of matrix $A$ are zero?Is this true that matrix$A$ is nilpotent?
0
votes
1answer
13 views

Diagonally dominant matrix for Cholesky?

I have a $10^6 \times 10^6$ dense SPD matrix, which I am called to invert, by using Cholesky factorization. However, I came across this statement: We start with the Cholesky and LU ...
1
vote
1answer
18 views

Find the standard matrix representation of the linear transformation T in M2,2

let $T: M_{2,2} \rightarrow M_{2,2}$ be a linear transformation defined by: $$T \left(\begin{bmatrix} a & b\\ c & d\\ \end{bmatrix}\right) = \begin{bmatrix}a + b& ...
0
votes
0answers
34 views

Ideals in the ring of $n\times n$ complex matrices [duplicate]

I want to find the left and right ideals in the ring of $n\times n$ complex matrices. Let's start with the left ideals: A subset $I$ of $R$ is called a left ideal of $R$ if it is an additive ...
1
vote
1answer
19 views

prove the following property related to singular value decomposition

Suppose $A$ is a $n\times n$ matrix. Show that the following are equivalent:(i), $A^2=BA$ for some non-singular $B$. (ii) $rank(A)=rank(A^2)$. (iii), $$Range(A)\bigcap Ker(A)=\{0\}$$, (iv) there ...
1
vote
1answer
29 views

If $A$ is negative-definite, then for a sufficiently big $k>0$ the eigenvalues of $M = kA + B$ are all with negative real part?

I want to prove the next statement: "If $A$ is a symmetric negative-definite matrix, then for a sufficiently big $k\in\mathbb{R}^+$, the eigenvalues of $M = kA + B$ are all with negative real part, ...
2
votes
0answers
44 views

How to calculate the eigenvalue of the following general matrix [duplicate]

Let the $n\times n$ matrix $Z$ with $(i,j)$-element defined by $Z_{i,j}=i+j$. How to calculate the eigenvalue of $Z.$? I have used Matlab to calculate it. I find no matter how bigger n is, there are ...
1
vote
1answer
35 views

Matrices $P$ such that $A$ is symmetric $\Longrightarrow $ $PAP^{-1}$ is symmetric

Let $M_n(\mathbb{R})$ be the (vector) space of all $n\times n$ matrices over $\mathbb{R}$. Let $Sym_n(\mathbb{R})$ denote the subspace of symmetric $n\times n$ matrices. $GL(n,\mathbb{R})$ acts on ...
0
votes
0answers
10 views

How to formally describe the lowest values of a vector / sorted vector?

I have a distance matrix D and would like to describe that I am just taking the mean (or median) of the 5 lowest values for each column. The programming implementation e.g. in R is fairly easy: ...
-1
votes
0answers
11 views

Question related to matrix and it's transpose.

Prove: For any matrices A and B and any scalars a and b, $(aA+bB)^t$ = a$A^t$ + b$B^t$.
2
votes
0answers
11 views

Eigenvalues and positivity of Hermitian Toeplitz matrices

I want to check the eigenvalues (and also the positivity) of the $n \times n$ complex Toeplitz matrix \begin{equation} T = \begin{bmatrix} r & z_1 & z_2 & z_3 &\cdots & z_{n-1}\\ ...
0
votes
0answers
15 views

real similar matrices [duplicate]

If real matrices $A$ and $B$ are similar to each other, prove that there is a real matrix $S$ such that $A=SBS^{-1}$. As we know, when $A$ and $B$ are similar to each other, then there exits complex ...