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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0
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1answer
19 views

Inverse of the sum of the inverse of 2 non-invertible matrices

Given that the following square matrices are non-invertible: $\bf A$, $\bf B$, and $\bf (A+B)$ and given that $\bf (A+I)$, $\bf (B+I)$, and $\bf [(A+I)+(B+I)]$ are invertible, is there a way to ...
0
votes
3answers
42 views

Does every invertible complex matrix have an eigenvector?

Over $\mathbb{C}$ does every invertible matrix have at least one non zero eigenvector?
1
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0answers
19 views

prove that a antisymmetric and invertible matrix is congruent to another matrix

let A be an antisymmetric and invertible matrix $A \in M_{2m}(\Bbb{R})$ prove that A is congruent to $$ \begin{pmatrix} 0 & I_m \\ -I_m & 0 \\ ...
1
vote
0answers
14 views

Eigenvalues of Certain Symmetric Block Matrix

What can we say about the relation between the eigenvalues of the following block matrix with identity diagonal blocks, and the singular values of the off-diagonal blocks: \begin{equation} ...
1
vote
2answers
26 views

All the cases for Image and Kernel

here alpha is a real variable, and I need to find the kernel and image for all values for alpha. Attempt: I can't seem to figure out all the cases which I need to evaluate, as the last two columns ...
0
votes
0answers
10 views

Interlacing Theorem on Singular Values

Does the Cauchy's interlacing theorem hold for "singular values" of matrices too? I saw on this publication first Theorem that it does. It states that singular values of a matrix interlace the ...
2
votes
2answers
20 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
3
votes
4answers
76 views

Is there always a matrix $X$ such that $X^2=A$?

Is it true that for every $A\in M_{2\times 2} (\mathbb{C})$ there's an $X\in M_{2\times 2} (\mathbb{C})$ such that $X^2=A$? For the matter of fact, I don't have a clue, other than evaluating the ...
0
votes
0answers
18 views

Proving that same solution set implies row equivalence

The question I am trying to solve is for a much simpler case: Suppose $R$ and $R'$ are $2\times3$ row-reduced echelon matrices and that the system $RX=0$ and $R'X=0$ have exactly the same ...
1
vote
0answers
12 views

How to Construct $N$-dimensional Unitary Matrix Basis

Galitski's Exploring Quantum Mechanics says on its page 29, (There are $N^2$ linearly) independent Hermitian matrices of rank $N$. The number of independent unitary matrices is also $N^2$, since ...
1
vote
2answers
41 views

Are those matrices congruent?

$$A = \left(\begin{array}{cccc} 1&0 \\ 0&-1 \end{array}\right), B=\left(\begin{array}{cccc} 1&0 \\ 0&2 \end{array}\right), C = \left(\begin{array}{cccc} 1&0 \\ 0&4 ...
0
votes
1answer
21 views

Can functions within a matrix adjust its size?

I've been working my way through a proof, and without going into the full extent of the details it's come down to whether a function G() exists such that the 1 by 3 matrix: ...
-3
votes
0answers
21 views

Need help on this matrix!! [on hold]

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
5 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 ...
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
30 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
7 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 & ...
1
vote
2answers
29 views

Regularity and invertibility of two parameterized 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
37 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 ...
1
vote
1answer
15 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}\\ ...
4
votes
3answers
118 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
9 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 ...
0
votes
0answers
9 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
1answer
24 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 ...
2
votes
2answers
38 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 ...
3
votes
4answers
178 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
2
votes
1answer
34 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
63 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} ...
7
votes
0answers
59 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 ...
2
votes
1answer
23 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 & ...
5
votes
0answers
38 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
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? ...
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
29 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 $B$ ...
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 ...
-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 ...
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 ...
1
vote
1answer
83 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 ...
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 ...
-2
votes
1answer
39 views

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

I'm new to matrices and I actually want to know how to calculate either the left or right inverse if the non square matrix possesses that. I'd also like to know about the pseudo inverse. I have just ...
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?
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 ...
1
vote
1answer
26 views

Different representations of a matrix in reduced row echelon form

EDIT: I decided to ask this question after working on this particular problem. I had the stupidity to think that row-reduced = row reduced echelon form. Brain fart, nothing more to see here... ...
0
votes
0answers
17 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}, ...
2
votes
1answer
41 views

Can you add a scalar to a matrix?

If I add a scalar to every element of a matrix, e.g. for a $2\times2$ matrix $$ \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix} + b \overset{?}{=} \begin{pmatrix}a_{11}+b & ...
0
votes
2answers
33 views

Final transformation matrix

I have a 3d object, to which I sequentially apply 3 4x4 transformation matrices, $A$, $B$, and $C$. To generalize, each transformation matrix is determined by the multiplication of a rotation matrix ...
1
vote
1answer
24 views

Matrix inequality $A^2 \succeq A$

If $A$ symmetric positive semidefinite matrix is the following inequality true. If $A \succeq I$ then \begin{align} A^2 & \succeq A \end{align} This is an equivalent of $a^2 \ge a$ is $a \ge ...
0
votes
1answer
28 views

3 Points in 3D Space to Develop an Arc or Circle

Background: I'm a Robotics Engineer and I am trying to develop a more flexible, modular, and robust program for our welding robots, which will minimize teaching time for new robots and also minimize ...