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
9 views

Need help on this matrix!!

So i got this question from my lecturer,and i am so dumbfounded in clarifying this question. In a football league,the price for every win,draw and lose is RM5000,RM3000 and RM1000 respectively.A team ...
1
vote
0answers
4 views

Condition number of positive definite matrix after rectangular orthogonal transformation on both sides

What is a lower bound on the condition number of $B A B^{T}$ (besides the trivial $\operatorname{cond}(B A B^{T}) \ge 1$) where $A$ is an $n \times n$ symmetric positive definite matrix, $B$ is a ...
4
votes
3answers
112 views

Can we prove that matrix multiplication by its inverse is commutative? [duplicate]

We know that $AA^{-1} = I$ and $A^{-1}A = I$, but is there a proof for the commutative property here? Or is this just the definition of invertibility?
0
votes
0answers
10 views

Regulary and invertibility of two paramterized matrices?

$$ C= \begin{bmatrix} 1+a & 2 & 3 & 4 & 5 \\ 1 & 2+a & 3 & 4 & 5 \\ 1 & 2 & 3+a & 4 & 5 \\ 1 & 2 & 3 & 4+a & 5 \\ 1 & 2 & ...
1
vote
1answer
21 views

Determinant of 5x5 matrices

Let A and B be 5x5 matrices with det(-3A)=4 and det(B^-1)=2. Find the det(A), det(B) and det(AB). My answer : det(A)=-12 , det(B)=1/2 and det(AB)=-6. Wish to check my answer, thank you.
1
vote
1answer
17 views

Determinant of 3x3 matrices

Let $A$ and $B$ be $3\times3$ matrices with $\det(A)=10$ and $\det(B)=12$. Find $\det(AB)$, $\det(A^4)$, $\det(2B)$, $\det((AB)^T)$. Answers: $\det(AB)=\det(A)\det(B)=120$ , $\det(A^4)=10000$ , ...
0
votes
0answers
6 views

Algortihm for Solving Linear equation from a Matrix

I have a set of linear equations from which I have built a matrix below: $M = \begin{bmatrix} p_1 g_1 & - \eta_1 p_2 g_2 & \cdots & - \eta_1 p_n g_n & s_1 & 0 & ...
10
votes
4answers
631 views

Is every noninvertible matrix a zero divisor?

Is every noninvertible matrix over a field a zero divisor? Related to this: What are sufficient conditions for a matrix to be a zero divisor over a noncommutative ring?
1
vote
1answer
33 views

What is $\frac{\partial x^TA^{-1}y}{\partial A}$?

I had trouble proving the following: If $ A\in \mathbb R^{n\times n}$ and $A$ is nonsingular, $x \in \mathbb R^{n\times 1}$, $y \in \mathbb R^{n\times 1}$, then $ \dfrac{\partial ...
3
votes
4answers
175 views

Matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ to a large power

Compute $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^{99}$ What is the easier way to do this other than multiplying the entire thing out? Thanks
1
vote
1answer
910 views

Least square solution based on the pseudoinverse solved efficiently with singular value decomposition

Hi apologies it's hard to type out the problem, I have a lecture slide on neural networks. It says the fitting error gives the matrix: N by M matrix of thi's multiplied by Mx1 weights minus Nx1 ...
1
vote
0answers
11 views

What can be said about a matrix with a constant column of ones with entries from a finite field?

I am working with matrices of the following structure: $A = \begin{pmatrix} 1&\alpha_{21}&\cdots&\alpha_{n1}\\ 1&\alpha_{22}&\cdots&\alpha_{n2}\\ ...
0
votes
0answers
6 views

Optimization problem involving semidefinite matrix variable that is constrained to be a tensor product

I would like to solve the following optimization problem. With scalar $R$ and nine (mutually orthogonal) $9$-dimensional column vectors $\vec v_i$ all given ($\vec v_i\!'$ is the row vector Hermitian ...
1
vote
2answers
40 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
0
votes
0answers
14 views

Find the minimum of a function for only positive values of the vector variable

Let variable vector $\vec{q}$ of size $m\times1$, and its diagonal counterpart $m\times m$ matrix $Q=diag(\vec{q})$, for some $m\in\mathbb{N}$. Define fixed parameter $n\times1$ vectors $\vec{p}, ...
0
votes
0answers
7 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
1
vote
2answers
525 views

Rotating the gradient

Suppose I have a triangle T in 3dimensional space and i want to rotate it in arbitrary ways. The coordinates for T are given by $f: T_R \in \mathbb{R}^2 \rightarrow T \in \mathbb{R}^3 $ where $T_R$ is ...
6
votes
2answers
74 views

Properties of matrices $M=UDU^*$ with $UU^*=Id$

I recently came across some matrices of the form $M=UDU^*$ (the superscript $*$ denotes the conjugate transpose), where $U \in \mathbb{C}^{r\times n}$ with $r<n$, $D \in \mathbb{C}^{n \times n}$ a ...
1
vote
1answer
222 views

Colleague Matrix

Can someone explain to me the concept of a Colleague Matrix. I tried to find some information online and I haven't been able to find anything. Example.. Given the function $$f (x) = x\bigg(x − ...
2
votes
2answers
36 views

The $\exp \circ \log$ function acts as the identity on unipotent matrices.

I am working through the exercises in "Lie Groups, Lie Algebras, and Representations" - Hall and can't complete exercise 9 of chapter 2 using the provided hint. In chapter 2, Hall defines the ...
1
vote
0answers
17 views

What is the derivative of the ReLu of a Matrix with respect to a matrix

I want to compute $\frac{\partial r(ZZ^tY)}{\partial Z}$ where the ReLu function is a nonlinear operator $r(x)=max(0,x)$ and $Z \in\mathbb{R}^{n\times m}$ is a matrix. I am wondering also if the ...
7
votes
0answers
58 views

Bound on the difference of two determinants

Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 ...
1
vote
1answer
44 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
2
votes
1answer
33 views

Suppose that $u,v \in \mathbb R^n$ with $u,v$ not equal to $\mathbf 0$, and let $A= I + uv^\top$.

a) Show that $1+v^\top u$ is an eigenvalue of $A$ and $u$ its eigenvector. b) Define the subspace $S$ of $\mathbb R^{n}$ to be $$S=\{x \in \mathbb R^{n}\mid v^\top x=0\}= \operatorname ...
1
vote
2answers
34 views

Does injective imply each $x$ matches to a unique $y$?

Injective means one-to-one matching, as in each $y$ is matched by only one $x$. However, does this mean that each $x$ matches only to one $y$?
1
vote
1answer
62 views

Computing $\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr)$

Suppose that $A \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A)\le n$. I want a lower bound on the following quantity $$\operatorname{Tr} ...
1
vote
1answer
28 views

Diagonalizing a matrix arising in a simple combinatorial situation

Maybe I'll return to this question a few hours from now and possibly even post an answer then. This concerns a matrix that I described in this answer. Start with a $\dbinom n2\times n$ matrix $B$ ...
2
votes
1answer
22 views

Determinant of a Certain 3 by 3 Block Matrix with Scaled Identity Blocks

What is the determinant or/and eigenvalues of the following 3 by 3 block matrix: $$\left[\begin{array}{ccc} \frac{3}{4}I & \frac{1}{4}I & \frac{1}{4}I \\ \frac{1}{4}I & \frac{3}{4}I & ...
4
votes
1answer
684 views

Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known

Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$. Let $D$ be a diagonal matrix with the same dimensions and rank. What is the fastest way to compute $(A+D)^{-1}$? ...
5
votes
0answers
35 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
3
votes
2answers
1k views

How to find closest positive definite matrix of non-symmetric matrix

I have a matrix A given and I want to find the matrix B which is closest to A in the frobenius norm and is positiv definite. B does not need to be symmetric. I found a lot of solutions if the input ...
0
votes
0answers
38 views

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
3
votes
1answer
44 views

Emil Artin on visualization of matrices

Someone called my attention to the fact that Emil Artin made very important remarks on the visual representation of matrices in some of his books. Could anyone tell me which precise book that is? ...
1
vote
1answer
75 views

Jordan canonical form of an upper triangular matrix

Find the Jordan canonical form of the matrix. Justify your answer. $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{bmatrix} $ My Try: The eigenvalues ...
2
votes
0answers
73 views

Best way to solve specific block-tridiagonal linear system (10000x10000 and larger)

To provide more context, this system came from energy balance equation on a mesh with (n,m) nodes in each direction. It's a linear system that looks like this (size of system in blocks n = 4, size of ...
0
votes
1answer
53 views

2x2 matrix multiplication issue

Let $$f_w(z)=z+w=\begin{bmatrix}1 & w \\ 0 & 1\end{bmatrix}z$$ where $z$ is a complex number. Shouldn't this be $w$ when $z=0$? However when I do the multiplication I get ...
0
votes
0answers
42 views

How following matrices equation is solved?

Suppose matrix $\mathbf{P}=[\mathbf{I_r} \mathbf{M}]$ and $\mathbf{Y}=\mathbf{G_t}\mathbf{P} =\mathbf{G_t}[\mathbf{I_r}\mathbf{M}]=[\mathbf{G_t}\mathbf{G_t}\mathbf{M}]$. if $\mathbf{G_t}$ has left ...
0
votes
1answer
19 views

Representing all pairs shortest path in a graph with a matrix

Given a graph $G(n,E)$ where $n$ is the number of nodes and $E$ represents the edges. Is there a way to represent or transform this into a matrix containing all the shortest paths between two pairs ...
-1
votes
0answers
22 views

Finding an algorithm to create a vector b given b*b' positive semi definite

My problem is the following: I have a column vector $b$, of positive or zeros values (at least one value should be $> 0$). I want $b b^T$ to be semi definite positive, and I want an algorithm ...
2
votes
3answers
62 views
+50

What is Homogeneous Coordinates? Why is it necessary in 2D transformation?

What is Homogeneous Coordinates? Why is it necessary in 2D transformation of objects in computer graphics? The concept of homogeneous coordinates in effect converts the 2D system a 3D one. So, why ...
-2
votes
1answer
36 views

Inverse of a non square matrix(left/right/pseudo/SVD) [on hold]

I'm new to matrices and I'd like to calculate the inverse of a non-square matrix, Say a $7\times 14$ matrix like this: $$\begin{bmatrix} 1& 4& 5& 8& 7& 5& 3& 7& 9& ...
6
votes
1answer
371 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
4
votes
2answers
50 views

$C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.

Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid ...
1
vote
1answer
72 views

Calculating the Dimension of a Subspace of $C\in Mat_{n\times n}(\mathbb R)$

Let $C\in \text{Mat}_{n\times n}(\mathbb R)$. Then which of the alternatives are correct: $\operatorname{dim}\langle I,C,C^2,\dots,C^{2n}\rangle$ is at most $2n$ $\operatorname{dim} \langle ...
1
vote
2answers
72 views
+50

Solving a matrix equation using numerical optimization

To my knowledge, if $A \in \mathbf{S}^n_{++}$, then given any $b \in \mathbb{R}^n$, the system of linear equations $Ax = b$ has a unique solution $x^* \in \mathbb{R}^n$. Moreover, the solution $x^* ...
1
vote
4answers
94 views

why does the reduced row echelon form have the same null space as the original matrix?

What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?
2
votes
2answers
68 views

If some power $A^n$ of a matrix $A$ is symmetric, is $A$ necessarily symmetric?

If $A^{n}$ is a symmetric matrix, should I conclude that A is also symmetric?
1
vote
1answer
76 views

Power of a matrix and its symmetricity

Let $A$ be a real $N\times N$ matrix. If $A^k$ is symmetric for some $k>0$, does that give away something about $A$.
0
votes
1answer
16 views

Adding a dependent row to a matrix with LI rows

Lets say my matrix is giving me a unique solution.What if I add another row that is some combination of already present rows?I know it would set the determinant to zero and now the solution may not ...
0
votes
0answers
18 views

Show that every Jordan matrix has a cyclic vector

Is my following reasoning correct? Since an $n\times n$ Jordan matrix has rank $n-1$ (because we can only make the last row the zero row), its geometric multiplicity is 1, which means the matrix has ...