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
27 views

Matrix/determinant inequality

I have studied the inequality that if $A-B$ is positive semi-definite, then $\det(A)\geq \det(B).$ I was trying to prove the other way around. That if we know that $A$ and $B$ psd and that ...
5
votes
1answer
125 views

Symmetric matrix eigenvalues

Let $A$ be an $n\times n$ matrix, with $A_{ij}=i+j$. Find the eigenvalues of $A$. A student that I tutored asked me this question, and beyond working out that there are 2 nonzero eigenvalues ...
3
votes
2answers
168 views

Row and column operations and matrix similarity

Take for example the following matrix: $$ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} $$ The elementary matrix equivalent to changing the first row ...
4
votes
1answer
53 views

If a matrix as well as its Hermitian part both have determinant one, must the matrix be Hermitian?

If $x\in\mathrm{M}_2(\mathbb{C})$, $y=\dfrac{x+x^{\dagger}}{2}$, and $z=\dfrac{z-z^{\dagger}}{2}$, then $x=y+z$. Also, $y$ and $z$ respectively are Hermitian and anti-Hermitian, i.e. $y^{\dagger}=y$ ...
3
votes
2answers
65 views

Infinite Matrices

Does anyone know how to prove that the set of all $K\times K$ column finite matrices over a ring $R$ $[\mathrm{CFM}(R)]$ forms a ring? I am also confused about the definition of multiplication in ...
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0answers
40 views

Maximum of norm

Given a matrix $A$ with $N$ rows and $d$ columns, I would like to prove (or disprove) the following. Let $q(f)=\|(\begin{pmatrix} ...
0
votes
2answers
27 views

What is meant by eigen spaces are non-orthogonal?

$M$ is a square matrix $M$ ( matrix representation of a linear operator $L$ acting on a hilbert space $H$ , $L: H \to H$ ) with eigen values $\lambda_i$ and corresponding eigen spaces $V_i$. I know ...
1
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2answers
33 views

Multiplying by a random matrix

Let $A\in \mathbb{Z^{m\times n}_q}$ be some matrix. Let $B\in \mathbb{Z^{n\times k}_q}$ be a random (uniform) matrix. Can I say that $C=AB$ is also random?
0
votes
1answer
93 views

Find the matrix $A$ such that the following is true:$ T_1(T_2(\mathbf{x})) = A\mathbf{x}$?

I found a matrix for $\mathbf{x} = (1, 1)$, and then took the $T_2$ transformation then the $T_1$, but it wasn't correct. I don't really understand what they're asking for me. Should I be finding ...
0
votes
2answers
28 views

Jordan canonical bases and form

Find the Jordan bases and the Jordan canonical form for the following matrices: $\begin{pmatrix} 2 & 3 \\ \\ 0 & 2 \end{pmatrix}$ (sorry about the formatting) So I found the eigenvalues ...
0
votes
1answer
64 views

Finite matrix groups as subgroups of $S_n$.

I have heard that all finite subgroups are isomorphic to a subgroup of $S_n$. I was thinking about examples of this. In particular I would like to know how this works for certain matrix groups. The ...
0
votes
2answers
373 views

Prove that an upper triangular matrix is invertible if and only if every diagonal entry is non-zero.

Prove that an upper triangular matrix is invertible if and only if every diagonal entry is non-zero. I have proved that if every diagonal entry is non-zero, then the matrix is invertible by showing we ...
0
votes
1answer
27 views

Relaxing the elements of a matrix

I try to understand a specific part of the paper "Consistent shape maps via semidefinite programming", where a binary symmetric Input matrix $X^{in}$ is given with $X^{in} \in \{0,1\}^{nm \times nm}$ ...
2
votes
1answer
48 views

Eigenvalues of $I-A^{t}A$ where $A$ is a semi orthogonal matrix

A friend asked me the following: Let $A$ be an $m\times n$ matrix over a field $\mathbb{F}$ s.t $AA^{t}=I_{m}$. Prove $I_{n}-A^{t}A$ is not invertible. My thoughts: If $m=n$ then $A$ is ...
0
votes
1answer
163 views

Difference between orthogonal complement and Gram-Schmidt process

Could someone explain what is the difference between these two? From my poor understanding they seem to do the same thing, given a set of vectors we find their corresponding orthogonal vectors. Maybe ...
1
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0answers
49 views

inverse of semi infinite toeplitz matrix

I have a semi infinite toeplitz matrix of the form $ A=\left(\begin{array}{ccccc} A_{0} & A_{1} & 0 & 0 & \cdots\\ A_{-1} & A_{0} & A_{1} & 0 & \cdots\\ 0 & A_{-1} ...
1
vote
1answer
52 views

How do I turn this Crank-Nicolson type equation into three vectors representing the middle, upper, and lower diagonals in a tridiagonal matrix?

I have the following homework problem: I have calculated the Crank-Nicolson equation to be Equation 1 $$ -200.05u_{m-1}^{n+1}+400.9995u_{m}^{n+1}-199.95u_{m+1}^{n+1} = ...
2
votes
0answers
50 views

Is the matrix with these coefficients invertible?

Let $0 \leq x_{i-1} < x_i < x_{i+1} \leq 1$. Let $p, q$ be functions that depend on that such that $p$ is positive and $q$ is non-negative. Let $c_i = a_{i+1,i} = a_{i,i+1}$. Let all other ...
12
votes
2answers
274 views

$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a ...
-1
votes
1answer
178 views

rational eigenvalues of integer matrix are integral

Let $A=A^T$ a real symmetric matrix with integer entries. How do you prove that a rational eigenvalue of A is integral?
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votes
2answers
231 views

Proving that $SL_2(\mathbb{Z}_5) / \{\pm I\}\simeq A_5$

I see here that one can prove that $$ SL_2(\mathbb{Z}_5) / \{\pm I\} \simeq A_5 $$ using the First Isomorphism Theorem. My question is how one would do that. I know that I need a surjective ...
1
vote
1answer
65 views

Vectors/matrices: How to show that $(v-w)(v-w)^t v = \frac{1}{2}\lVert v-w\rVert^2 (v-w)\quad$ (vector notation)

Given the following details: $v \neq w$ two vectors of $\mathbb{R}^n$ with $\lVert v\rVert = \lVert w\rVert$. Let $u = \frac{1}{\lVert v-w\rVert}(v-w)$ and $H = I - 2uu^t$. Suppose $x$ is an ...
2
votes
1answer
34 views

Why are these matrix row operations even allowed simultanously on more than one matrix?

My Differential Equations book is going over finding the inverse of Matrices, and clearly I've forgotten my college algebra. I have no idea why this works. The first example gives this: ...
1
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0answers
66 views

SVD of a block partitioned matrix

Given a block partitioned matrix $\boldsymbol{A}$ $$ \boldsymbol{A} = \begin{bmatrix} \boldsymbol{A}_{1,1} & \boldsymbol{A}_{1,2} & \cdots \\ \boldsymbol{A}_{2,1} & ...
0
votes
1answer
41 views

Matrix multiplication with complex numbers

I am trying to simplify the following inequality \begin{align*} &\log(1+\mathbf{h}_2^H \mathbf{S} \mathbf{h}_2) \leq \log(1+\mathbf{h}_1^H \mathbf{S} \mathbf{h}_1) \tag1 \end{align*} ...
2
votes
3answers
524 views

Does matrix addition give you a matrix or a number?

I am very confused by something our lecturer said today: We were given two matrices: $B=\begin{pmatrix}2 & 3\\ 2 &0 \\ 0&3\end{pmatrix}$ C=$\begin{pmatrix}6 ...
0
votes
1answer
43 views

identity operator, direct sums, and projections

Let W be finite-dimensional vector space. Let $P: W\to W$ be a projection. Let U = Range(P) and V=Ker(P) (a) show that P is the identity operator on U. I dont understand the problem ...
2
votes
1answer
153 views

In general find: eigenvalues and ''-vectors of symmetric matrices with all rowsums equal | Specific example (where all non-diag. elements equal)

If you have a situation where every row-sum is equal in a matrix A, this sum equals one of the eigenvalues of the matrix. It might make a difference in cases where the matrix is symmetrical? Using ...
3
votes
1answer
56 views

The Classification of almost complex structures (almost) tamed by a quadratic form

Preamble I am currently dealing with the almost complex structures associated with a non-degenerate quadratic form. In particular, I am interested the size of the almost complex structures as a ...
0
votes
1answer
138 views

Finding a pair of Orthogonal Vectors

Want: Pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3) My attempt at a solution: I got stuck...
2
votes
1answer
55 views

What kind of matrix norm satisifies $\text {norm} (A*B)\leq \text {norm} (A)*\text {norm} (B)$ in which A is square?

$||A\times B||\le ||A||\cdot ||B||$ is not always correct. But which kind of matrix norm satisifies this formula for square matrix $A$ and arbitrary matrix $B$?
8
votes
1answer
87 views

Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
1
vote
3answers
84 views

Volume of a parallelepiped when not given values for three vectors

There is a parallelepiped determined by three dimensional vectors x, y, and z. The volume of this parallelepiped is $11$. What is the volume of the parallelepiped determined by the three dimension ...
1
vote
2answers
49 views

Find a basis for the row space and a basis for the column space

By inspection, find a basis for the row space and a basis for the column space for the following matrix: $$ \begin{bmatrix} 1 &2& -2& 7 \\ 0 &1& 3 &5 \\ 0& 0& 1 ...
5
votes
2answers
127 views

Degree of minimum polynomial at most n without Cayley-Hamilton?

Let $T$ be a linear transformation of an $n$-dimensional vector space $V$ over a field $k$. It's pretty easy to define the minimum polynomial of $T$ and make sure its degree is between $1$ and $n^2$, ...
2
votes
0answers
50 views

Matrix Point-Wise Vector Multiplication

I have the following equation, where $\mathbf{M}$ denotes a singular Square Matrix (dim= $n$ x $n$), $\mathbf{x}$ and $\mathbf{y}$ denote vectors (with dimension $n$, too). The operator $\odot$ ...
1
vote
2answers
47 views

What is the relationship between dimension of eigen space and multiplity of eigen value?

Is there a relationship between dimension of eigen space with respect to an eigen value $\lambda_i$ and multiplicity of eigen value $\lambda_i$ ( by multiplicity I mean if $(\lambda-2 )^3(\lambda ...
2
votes
2answers
264 views

how to combine angle rotations along different axes into one rotation along a single vector [duplicate]

So, lets say I have some rotation a about the x-axis(vector:$(1, 0 ,0)$) and some other rotation about y-axis(vector $(0, 1, 0)$) and a rotation about the z-axis(vector: $(0,0,1)$). How would I ...
0
votes
1answer
104 views

Orthogonal projection matrix P onto the range of a 3x2 matrix

I have a 3x2 matrix A = {{1,-1},{2,-1},{3,1}}. I need to find the orthogonal projection matrix P onto the range of A. I know that the orthogonal projection is the outer-product / inner-product, that ...
1
vote
1answer
23 views

Polynomial Factorisation - Linear Algebra

Im attempting a linear algebra question in which I have been given the following quadratic form $q(x,y,z) = x^2+25y^2+10xy+2yz$. I have to find a basis $B$ such that $[f]_B$ has the real canonical ...
2
votes
0answers
79 views

How to compute determinant (or eigenvalues) of this matrix?

Let us have the $n \times n$ circulant matrix given by \begin{equation} C(c_0,c_1,\cdots, c_{n-1}) =\begin{bmatrix} c_0 & c_1 & c_2 &\cdots & c_{n-1}\\ c_{n-1} & c_0 & c_1 ...
4
votes
0answers
234 views

Conditions for Trace Inequality Tr( ( A² - B² ) Z) >= 0

Consider the $M \times M$ complex positive semidefinite matrices ${\bf A}, {\bf B}, {\bf Z}$. We have the relation $\mu_{\text{max}}{\bf I} \succeq {\bf A} \succeq {\bf B} \succ \mu_{\text{min}}{\bf ...
0
votes
1answer
92 views

what does it by raising a matrix to the power of $1/2$?

I came across the following which I did not understand at all. Let $A$ be a positive semi-definite. If $A(I-B)$ is positive definite, then the eigenvalues of $$A^{1/2}(I-B)A^{-1/2} = I ...
2
votes
1answer
91 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 ...
1
vote
2answers
39 views

Simple question: How do these changes affect the determinant of my matrix?

I assume there's some simple rule to follow, but I can't seem to see what it is. Given $$\det \left[\begin{array}{ccc} a &1 &d\cr b &1 &e\cr c &1 &f\cr \end{array}\right] = ...
0
votes
1answer
28 views

Let $M =\{ f(x)\in P_3 | \int_0^1f(x)dx = 0\}$ Find basis for M.

Let $M =\{ f(x)\in P_3 | \int_0^1f(x)dx = 0\}$ Find basis for M. solution: $P_3$ is the set of all polynomials of degree strictly less than 3, ($f(x) = a_2x^2+a_1x+a_0$). hence, $\int_0^1f(x)dx = ...
0
votes
1answer
175 views

Rotation Matrix with (Cos(theta) = 0,Sin(theta) = 1) as Identity

In my program, all rotations are handled with unit-vector orientations: $$[x,y] = [\cos{\theta}, \sin{\theta}]$$ In the game engine I'm using for visualization (Unity3D), the $Y$ axis is forwards - ...
0
votes
0answers
30 views

Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace.

assume K, L are proper subspaces. Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace. Solution: if $v_1,v_2\in K$, then $c_1v_1+c_2v_2 \ in K$ [because K is a subspace] if ...
0
votes
2answers
79 views

Prove that if u and v are vectors in $\mathbb{R}^n$, then $\langle u,v\rangle =1/4\|u+v\|^2-(1/4\|u-v\|^2)$

I seem to always have troubles when starting proofs. My professor said that the proofs he gave us today are mostly one line proofs, but I just don't know where to start with this one. What I've ...
-1
votes
1answer
83 views

Finding a pair of orthogonal vectors in $R^4$

Find a pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3). What i have tried so far: