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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7 views

If all the columns of a linear system are pivot columns, then it has a unique solution.(Proof Verification))

I found the following theorem(CSRN) on the website http://linear.ups.edu/html/theorems.html . I am only interested in proving the r = n part. Theorem: Suppose A is the augmented matrix of a ...
2
votes
4answers
101 views

Matrix 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
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1answer
30 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
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2answers
33 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
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1answer
53 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} ...
3
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0answers
42 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
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1answer
15 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 & ...
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0answers
32 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
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1answer
41 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
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1answer
59 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 ...
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1answer
27 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
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1answer
50 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
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0answers
21 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
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0answers
40 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
69 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
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2answers
47 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
34 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& ...
1
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4answers
92 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
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2answers
66 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?
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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
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1answer
24 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... ...
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0answers
5 views

Minimize real valued scalar function $f(Q)$, where $Q=diag(\vec{q})$ subject to $q_j\geq0$ $\forall j\in\{1,2,…,m\}$ (positive vector)

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
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1answer
40 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
32 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
27 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 ...
0
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4answers
75 views

Prove that $g(A)$ is an invertible matrix

Let $A\in M_n(\mathbb{C})$ and let $\lambda\in\mathbb{C}$. Prove that if $\lambda$ is not an eigenvalue of $A$ then $A-\lambda I$ is invertible. Moreover, for $g(x)\in \mathbb{C}[x]$, prove that if ...
15
votes
4answers
1k views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
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votes
1answer
22 views

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $? [on hold]

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $ ?
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0answers
14 views

What does $M_{uv}^l$ represent?

Let $M$ be any non negative square matrix. What does $M_{uv}^l$ represent? $M_{uv}^l$: $uv$ entry of $M^l$. (When $A$ is adjacency matrix of a graph, then $A_{uv}^l$ is number of walks of length $l$ ...
0
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1answer
18 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 ...
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0answers
26 views

Matrix polynomial [on hold]

Suppose: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a ...
0
votes
1answer
22 views

Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
4
votes
4answers
46 views

Equivalent definitions of an orthogonal matrix.

I wish to show that the following definitions of an $n \times n$ real matrix $Q$ are equivalent: $QQ^T=I$, $Qx\cdot Qx=x\cdot x$ for all $x\in \mathbb{R}^n$. I found it easy to show that ...
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0answers
13 views

E as an expectation of a quadratic form [on hold]

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
0
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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 ...
1
vote
2answers
36 views

Technical question in Vandermonde determinat proof

I can follow the proof given in (2nd proof, or the induction proof), until the sentence: "From the Expansion Theorem for Determinants‎, we can see that the coefficient of $x_k$ is:". I don't ...
2
votes
1answer
25 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
4
votes
2answers
78 views

Is this matrix invertible?

I have been working on a proof and am stuck with showing that the below matrix is invertible. I am not interested in the explicit inverse, only showing it has a nonzero determinant as the existence of ...
1
vote
2answers
25 views

spectral radius

Does the spectral radius of a matrix defines a norm? I mean does it satisfy the properties of norm, ie. $$||x|| \ge 0$$ $$||x|| = 0 \implies x=0$$ $$||kx|| = |k|\;||x||$$ $$||x+y||\le ||x||+||y||$$
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votes
3answers
30 views

Find the matrix product when possible

$\begin{bmatrix}-1& 3\\ 3 & 4 \end{bmatrix} * \begin{bmatrix}0& -2 & 4\\ 1 & -3 & 2\end{bmatrix} $ I realized that there is no third matrix column so does that mean I assume ...
1
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2answers
48 views

Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta ...
0
votes
4answers
35 views

Find the indicated matrix

$$ A=\begin{bmatrix} 2& 3\\ 2 & 4 \end{bmatrix} $$ $$ B=\begin{bmatrix} 0& 4\\ -1 & 6 \end{bmatrix} $$ Find 2A+B Would I go 2*2+2*3+2*2+ *4+ 0+4+-1+6?
0
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0answers
37 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 ...
0
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0answers
25 views

Can I do Gaussian Elimination on this? (mod 2)

I have this matrix in GF(2): [0, 0, 1, 0] [1, 1, 0, 0] [0, 0, 0, 1] It's not a square matrix but I tried to do Gaussian elimination on it anyway after adding a ...
2
votes
1answer
59 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
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1answer
19 views

What is the characteristic polynomial of power of a matrix

If the c.p. of A is $(\lambda-2)^3(\lambda+2)^2(\lambda+3)$, how can I find the c.p. of $A^2$? Would it be $(\lambda-4)^3(\lambda+4)^2(\lambda+9)$? Thanks!
0
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1answer
37 views

Perform the operation or operations when possible.

$\begin{bmatrix}-5& -9\\ 9 & 3 \end{bmatrix} + \begin{bmatrix}8& 5\\ -4 & -1\end{bmatrix} -\begin{bmatrix} 4& -7\\ -9 & -6 \end{bmatrix} $ Also, I was trying to add ...
3
votes
4answers
63 views

Does an $n\times n$ matrix $A$ only have an inverse if $rank(A)=n$? If so, why?

I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example ...
6
votes
2answers
38 views

minimum eigenvalue for difference of two matrices

Let $A$ a symmetric positive definite matrix, and $B$ a matrix constructed from $A$ by setting all its off-diagonal elements to zero. Then is there a way to see for which values of positive scalars ...