Tagged Questions

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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2
votes
1answer
37 views

How to decompose a matrix into an antisymmetric matrix plus a multiple of the identity

I was given a problem to solve earlier that I couldn't figure out. I don't still have it, but it was basically: Given the invertible matrix $A$, find the invertible matrix $P$, such that ...
3
votes
1answer
44 views

Bilinear functional to be elementary

Let $V$ be a $n$ dimensional vector space over $\Bbb C$. We say a functional $f:V\times V\to \Bbb C$ is bilinear if $f$ is linear in each variable if the other variable is fixed. And $$f$$ is called ...
0
votes
3answers
29 views

A quick way to generate 3x3 matrices with determinant equal to 1?

Perhaps a formula involving the row number and column number of an element or just some parametric equations for each element. I know that I can just multiply two of these matrices together to get ...
6
votes
2answers
81 views

Proving if A and B are matrices such that A, B, and AB are normal, then BA is also normal.

If A and B are matrices such that A, B, and AB are normal, then BA is also normal. I've seen this statement around, although I've only seen the site/publication/etc... state that it was proven by ...
-1
votes
4answers
44 views

True or false? Prove it.

If $A$ is an $n\times n$ invertible matrix and $B$ is an $n\times m$ matrix, then $\operatorname{rank}(AB) = \operatorname{rank}(B)$. Is this true or false? I've tried proven that if $B=0$, then ...
0
votes
0answers
19 views

Find the values of x,y,z so that the 3 x 3 matrix is singular?

Find the values of x, y, z that the matrix is singular? With an explanation.
2
votes
2answers
29 views

Use the cayley hamilton theorum to work out high powers of matrices

Let Matrix $$A= \left( \begin{array}{ccc} 1 & 2& 3 \\ 0 & 1 & 0 \\ 0 & 5 & -1 \end{array} \right) $$ Compute $A^{25}$ using the cayley hamilton theorum I know i use ...
0
votes
3answers
25 views

Determine if a set is linearly independent or dependent.

If $S = \{r,u,d\}$ and $S$ is a set of linearly independent vectors. and if $x = r + 4u + 2d$, determine whether $T = \{r,u,x\}$ is a linearly independent set as well. Not sure how to go about solving ...
0
votes
1answer
12 views

Why use transpose in finding if a subset is also a subspace.

I had a homework question in my linear algebra course that asks: Are the symmetric 3x3 matrices a subspace of R^3x3? The answer goes on to prove that if A^t = A and B^t = B then (A+B)^t = A^t + B^t ...
-1
votes
0answers
18 views

Upper bound on Lyapunov equation solution

We know from literature on the Lyapunov equation that there exists a unique, symmetric, positive definite solution for the matrix $P$ in the equation $$A'P+PA=-I,$$ where $A$ is a Hurwitz matrix. Is ...
2
votes
1answer
47 views

Embed $1$-dimensional torus in $SO(2)$

Let $k$ be an algebraically closed field, and let $k^*$ be the one dimesional torus. We want to embed it in $SO(2)$ , the group of matrices $A$ such that $\det A=1$ and $A^tA=Id$. My first attempt ...
0
votes
0answers
3 views

Normalizing points before computing covariance - what impact on covariance entries?

Let $x_1,\dots,x_n$ be vectors in $\mathbb{R}^p$. Let $T$ be some positive-definite matrix. Can we say something concrete about the entries of $$A = \sum_{i=1}^n \frac{xx^T}{x^T T^{-1}x}?$$ Notice ...
0
votes
1answer
31 views

How to prove distributive property of a determinant?

How to prove that $|A\cdot B| = |A|\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.
0
votes
0answers
22 views

Quadratic Sieve, matrix problem

I read this: Quadratic Sieve Matrix Reduction and I am basically stuck. My Gaussian elimination says the answer is v= 0,0,0. Although you can clearly see that the correct answer is (1,1,1). How does ...
1
vote
3answers
196 views

Sum of eigenvalues of a symmetric matrix

Problem to calculate the sum of eigenvalues of a matrix: $$ \begin{pmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \\ \end{pmatrix}$$ I can ...
0
votes
1answer
30 views

Norm of Triangle Matrix

How to find the norm of the following matrix, please? Thank you! $$T := \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix},$$ and $$\|T\| = \sqrt{n^2+1}.$$
1
vote
2answers
43 views

Prove the matrix $ \left( \begin{array}{ccc} B & A^T \\ A & 0 \\ \end{array} \right)\ $ is nonsingular [on hold]

Suppose the matrix $A\in\mathbb{R}^{m\times n}$, $m\leq n$, and has full row rank $m$, $B\in\mathbb{R}^{n\times n}$ is a symmetric, $Z\in\mathbb{R}^{n\times(n-m)}$ is the matrix whose columns span ...
1
vote
2answers
24 views

Finding the $\lim_{n\to\infty}(A^{n})$ of a complex matrix

Let $$ A = .5\begin{pmatrix} 1+\alpha & -1+\alpha\\ -1+\alpha & 1+\alpha\end{pmatrix}. $$ where $\alpha$ is a complex number. For which values of alpha does the limit ...
0
votes
1answer
14 views

$A\in\mathcal{S}^+_n,A={}^tMM$

How can I prove that any symmetric positive matrix $A\in\mathcal{S}^+_n(\mathbb{R})$ can be written $A={}^tMM$ where $M$ is an invertible matrix ? This is probably a duplicate, but I have not been ...
0
votes
5answers
99 views

Determine $\lim_{n\to\infty}A^{n}$

For matrix $$ A = \begin{pmatrix} 7/5 & 1/5\\ -1 & 1/2\\\end{pmatrix}. $$ Determine $\lim\limits_{n\to\infty} A^{n} $ Is the limit related to the eigenvalues? Using Matlab it appears that ...
2
votes
1answer
31 views

How to take the derivative of Matrices

I was browsing the derivation of the Least Squares estimates and stumbled about this problem. It said that: $$E = (Y + XB)^2$$ $$\frac{dE}{dB} = -X^TY + X^TXB$$ It is to my understanding that the ...
0
votes
3answers
48 views

Unit Eigenvalue if Determinant of an Orthogonal matrix is 1 [on hold]

For a (2n+1)x(2n+1) orthogonal matrix M, det(M)=1. Show M has a unit eigenvalue.
0
votes
3answers
73 views

Can product of two singular matrices be invertible?

Suppose $A,B$ are square matrices of size $n\times n$. Can $AB$ be invertible, even though both $A$ and $B$ are singular (not invertible)? And if not, does it follow that if $A_1 \times A_2 \times ...
0
votes
0answers
9 views

Representing the sum of squared diagonal elements of a matrix with trace function

Assume that we have a $n\times n$ vector called $A$. I am interested in computing $\sum_{i=1}^n A_{ii}^2$ (i.e., sum of the squared diagonal elements). However, I want to do so as trace function and ...
0
votes
3answers
38 views

How to check that this is an orthogonal linear map with $\det (A) = 1$, so it is a rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
0
votes
1answer
5 views

Calculate 1-norm of a vector using another matrix or vector

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply ...
0
votes
0answers
21 views

How to find the axis of the rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
0
votes
2answers
23 views

Calculate absolute value using matrix

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply ...
0
votes
1answer
14 views

Limit of a defined positive symmetry

Let $A:\mathbb{R}\rightarrow S_n^{++}(\mathbb{R})$ continuous with a continuous derivative. We suppose that $\forall t\in\mathbb{R},\mathrm{tr}(A'(t))\le-\mathrm{tr}(A(t))$. Show that ...
0
votes
0answers
32 views

Is there an efficient way to check if a matrix is a circulant under some simultaneous permutation of rows and columns?

I am interested in both practical and theoretical answers to this. I have a 16x16 matrix with only three different element values. The diagonal entries all have the same value. I would like to know ...
0
votes
2answers
13 views

Coordinate/matrix multiplication

If I multiply a 3d coordinate (padded with a 1 to make it a 4x1 matrix) with a transformation matrix, I get a 1x4 matrix which contains my new (transformed) 3d coordinate. Knowing that matrices must ...
2
votes
1answer
184 views

Show that SO(n) is a normal subgroup of O(n)

Show that SO(n) is a normal subgroup of O(n). A normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. SO(n) is the set of orthogonal ...
0
votes
1answer
17 views

Row space and column space intution

I have two questions. I know if I multiply a matrix like $A$ by vector $x$ then $Ax$ is like linear combination of columns of $A$. Now I would like to know what is the intuition when I multiply a ...
2
votes
0answers
20 views

What is the simplest way to solve determinant of a $n \times n$ matrix by upper and lower triangular matrices?

I know the basic rules to solve for the determinant of an $n \times n$ matrix using upper and lower triangular matrices, but what is the simplest way?
0
votes
1answer
30 views

Wronskian of a fundamental set of solutions

(instead of the dot above, i used ' and ", am I correct in thinking that these are equivalent?) Consider the system of equations, $$x'_1=x_2$$ $$x'_2=-q(t)x_1-p(t)x_2$$ where $q(t)$ and $p(t)$ are ...
1
vote
2answers
15 views

Hermitian Matrix is complex matrix

If M is an nxn complex matrix with $M^T = M^*$, then M is Hermitian. So, I know that an nxn complex matrix is called hermitian iff $M^H$=M. And $(M^T)^* = (M^*)^T = M^H$, so does this imply that the ...
3
votes
3answers
50 views

Prove that for a real matrix $A$, $\ker(A) = \ker(A^TA)$

So clearly the kernel of $A$ is contained within the kernel of $A^TA$, since $$A^T(A\vec{x}) = \vec{0} \Rightarrow A^T(\vec{0}) = \vec{0}$$. Now we need to show that the kernel of $A^TA$ is contained ...
0
votes
1answer
19 views

Given a homogeneous system, what can we say about a similar but nonhomogeneous system?

Alright, so we have a homogeneous system of 8 equations in 10 variables (an 8 x 10 matrix, let's call it A). We have found two solutions that are not multiples of each other (lets call them a and b), ...
3
votes
2answers
25 views

$A^TCA \leq B^TCB \Rightarrow A^TA \leq B^TB$?

Let $A$,$B \in \mathbb{R}^n$, $C\in\mathbb{R}^{n\times n}$, and $C=D^TD$ where $D$ is a $n\times n$ psd matrix, is it guaranteed that $A^TCA \leq B^TCB \Rightarrow A^TA \leq B^TB$?
2
votes
1answer
35 views

Trace of a matrix

"the preceding 2 scalar solutions correspond to the vector solutions $x^1(t)=(t,1)^T$ and $x^2(t)=(t^2,2t)^T$ which have the Wronskian $$W(t)=\det\left[\begin{array}{lr} \mbox t & t^2\\ \mbox 1 ...
1
vote
1answer
38 views

Diagonalization of a Matrix in terms of other matrices and eigenvalue

Task: Let A be a symmetric matrix having only one eigenvalue λ and C be a matrix that diagonalizes A by a similarity transformation. Find a simplified expression for A in terms of λ, C, and I, the ...
1
vote
0answers
15 views
2
votes
2answers
25 views

Find bases for eigenspaces of A

$$A = \begin{pmatrix} 6 & 4 \\ -3 & -1\end{pmatrix}$$ Find the bases for eigenspaces $E_{\lambda_1}$ and $E_{\lambda_2}$ of $A$. I don't really know where to start on this problem.
0
votes
2answers
13 views

Do elementary row operations give a similar matrix transformation?

So we define two matrices $A,B$ to be similar if there exists an invertible square matrix $P$ such that $AP=PB$. I was wondering if $A,B$ are related via elementary row operations (say, they are ...
5
votes
2answers
59 views

Maximum dimension of a nilpotent vector space

What is the maximum dimension of a vector space of $\mathcal{M}_n(\mathbb{R})$ containing only nilpotent matrices ? ($\mathcal{M}_n(\mathbb{R})$ : matrices $n\times n$ with coefficients in ...
0
votes
0answers
9 views

Order finding (modular) - Kitaev's Factoring Algorithm

Show Ma is Reversible and Unitary. This is the solution I have found. I understand the proof for the most part, however I don't think it is right. If it isn't right to prove Ma is Reversible by ...
1
vote
1answer
19 views

$M_n$ equivalent to the $C^*$-algebra generated by Jordan Block?

Suppose $J$ is the $n\times n$ Jordan Block matrix with all zero eigenvalues, (a.k.a. the "shift" matrix) $\begin{pmatrix} ...
0
votes
1answer
25 views

Diagonalizing zero matrix

Consider the matrix $A = 0$ that is diagonalized by the matrix $$S = \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}.$$ What is the diagonal matrix? I'm confused because I thought you could ...
0
votes
0answers
19 views

Fast inversion of triangular matrix

I wrote the code below to invert an upper triangular matrix, avoiding any possible multiplication/subtraction by zero. It just uses $\frac{1}{6}n^3+\ldots$ flops instead of $n^3+\ldots$ flops. ...
0
votes
2answers
28 views

Left ideals of matrix rings are direct sum of column spaces?

Let $\mathbb K$ be a field and $M_n(\mathbb K)$ be the ring of the $n\times n$ matrices with entries in $\mathbb K$. Let $C_j\subset M_n(\mathbb K)$ be the subspace of all matrices which have all ...