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
28 views

How to find all monomials $\left\{\left.x^n\in P_m\right|T(x^n)=0\right\}$ and which are in $\text{ker }T$?

Let $P_5$ be the set of one variable polynomials with real coefficients, whose degree are $\leq5$. Let $T$ be a linear transformation ...
1
vote
1answer
153 views

What does norm of a matrix mean?

I was reading the proof of SVD decomposition form here SVD decomposition proof. I was able to follow the proof except for one thing, they define norm of a matrix as $$|A|_2= \text{sup}_{v_1 \in ...
0
votes
1answer
55 views

How can the homogeneous system AX=0 have infinatly many solutions if |A|=0

let $A \in M_{nxn}$ & $|A|$ be the determinant of the matrix $A$ Why does the homogeneous system \begin{equation*} AX=0 \end{equation*} Have infinitely many solutions when: $|A| = 0?$
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votes
2answers
29 views

If $A$ has exactly $k$ nonzero eigenvalue and $A$ is normal, why is $\operatorname {rank}(A)=k$?

If $A \in {M_n}$ has exactly $k$ nonzero eigenvalue and $A$ is normal, why is $\operatorname {rank}(A)=k$?
2
votes
2answers
78 views

How find $a,b\in\mathbb R$ if are given the matrices $AB$ and $BA$?

Question: Let $A_{4\times 3},B_{3\times 4}$ be real matrices such that $$BA=\begin{bmatrix} -9&-20&-35\\ 2&5&7\\ 2&4&8 \end{bmatrix},\ AB=\begin{bmatrix} ...
2
votes
1answer
74 views

If $A$ is a $k \times k$ submatrix of $n \times n$ unitary matrix and $2k>n$. why that some singular value of $A$ is equal to $1$

If $A$ is a $k \times k$ submatrix of $n \times n$ unitary matrix and $2k>n$. why does some singular value of $A$ is equal to $1$?
0
votes
1answer
14 views

Existence of a colum matrix such that norm of the colum matrix after operated on by a matrix is one?

For any given matrix $A$ ( $m$ x $n$ ) is it possible to find a $n$ x $1$ column matrix such that $$|Av| >0, \; \text{where }\; |v|=1$$ Here $|\; |$ stands for norm. ...
3
votes
1answer
263 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 ...
2
votes
1answer
310 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
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1answer
53 views

$X$ is normal matrix and $AX=XB$ and $XA=BX$.why $A{X^*} = {X^*}B$ and ${X^*}A = B{X^*}$?

Let $A,B,X \in {M_n}$ and $X$ is normal matrix and $AX=XB$ $XA=BX$ Why $A{X^*} = {X^*}B$ and ${X^*}A = B{X^*}$?
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2answers
507 views

If $Ax = b$ has more than one solution so does $Ax = 0$, where $A$ is $m\times n$ real matrix.

Problem: If $Ax = b$ has more than one solution so does $Ax = 0$, where $A$ is $m\times n$ real matrix. In the explanation part it is written that when $Ax = b$ is consistent the solution sets of ...
-1
votes
1answer
146 views

Symmetric matrix with with all positive/zero elements. How to ensure it is PSD?

I have the following matrix A: symmetric all positive and/or zero values the main diagonal is all the same variable, x. To ensure that the matrix A, is PSD, must I only ensure that x>=0? It seems ...
0
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0answers
52 views

Calculation the Correlation Matrix of a Random Observation Vector

I've googled and looked around on StackExchange but I can't quite find the answer. I'm trying to calculate the correlation matrix R. Below I've included a screenshot of some text that explains how. ...
2
votes
0answers
91 views

Efficient computation of matrix determinant in finite ring

I am trying to implement generalization of Hill cipher. My idea is very simple: the size of key matrix should be arbitrary number not only three. All steps of this cipher is trivial except computation ...
2
votes
0answers
45 views

Smallest bound for convex combination of columns of non-negative matrix

The problem can be formulated as following linear program: $\min_{\mathbf{x},y}\;\;y$ subject to: $\mathbf{Ax}\le y\mathbf{1}$ $\mathbf{x}^T\mathbf{1}=1$ and $x_i \ge 0,\;\forall i$ Here, ...
13
votes
3answers
6k views

why determinant is volume of parallelepiped in any dimensions

for $n = 2,$ I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculate the area by coordinates. But how can one easily realize that it is true ...
4
votes
3answers
122 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 ...
2
votes
1answer
89 views

Eigen value of principal submatrix.

I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the ...
0
votes
1answer
67 views

“distance” metric between two bases modulo determinant, rotation and chirality

I'd like some kind of metric that tells me how similar two complete, not necessarily orthonormal bases (represented by non-singular matrices $B_1, B_2 \in \mathbb{R}^{n \times n}$) are to each other, ...
1
vote
1answer
216 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 ...
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vote
3answers
41 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 ...
2
votes
2answers
435 views

Strassen Multiplication?

How are the values of the 7 new matrices derived? I'm referring to the values that reduce matrix multiplication to 7 multiplications per level: $M_1 = \left(A_{1,1} + A_{2,2}\right)\left(B_{1,1} + ...
3
votes
1answer
77 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 $A$ ...
0
votes
0answers
61 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 ...
3
votes
0answers
79 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} ...
0
votes
3answers
91 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
47 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) ...
3
votes
2answers
50 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 ${A^{ - 1}}$?
2
votes
1answer
27 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
2answers
61 views

How to find eigenvalues of this matrix

How to find eigenvalues of this matrix: $\left( \begin{array}{ c c } 2 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & -1 & 2 \end{array} \right) $ ATTEMPT: $(2-λ) ...
0
votes
1answer
138 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)} ...
0
votes
0answers
30 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
votes
0answers
27 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?
2
votes
1answer
86 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 ...
5
votes
1answer
133 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 $D1$ ...
14
votes
3answers
2k views

Traces of all positive powers of a matrix are zero implies it is nilpotent

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be 0, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like to ...
2
votes
1answer
108 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? ...
0
votes
1answer
23 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
228 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 ...
1
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0answers
13 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 ...
0
votes
0answers
32 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 ...
0
votes
2answers
29 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
95 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
33 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
0
votes
1answer
27 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 $(A,B^{−1})$ ...
1
vote
1answer
73 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
1answer
35 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
43 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 ...
1
vote
1answer
26 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 ...
0
votes
1answer
40 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 ...