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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4
votes
1answer
104 views

Why does $ (A^T x) · y = x · (A y) $ hold?

Why does $ (A^T x)· y = x ·(A y) $ hold? The proof has to do with properties of transposes. I did a proof using coordinates (which was correct) but there is an infinitely easier way to do it. A is ...
7
votes
3answers
128 views

Let $A$ be a $n× n$ real matrix with $A^2 = A^T$. Show that every real eigenvalue of $A$ is either $0$ or $1$.

Let $A$ be a $n×n$ real matrix with $A^2 = A^T$. Show that every real eigenvalue of $A$ is either $0$ or $1$. My thoughts: $A^2 = A^T$ $\implies$ $A.A=A^T$ $\implies$$(A.A)^T=A$ $\implies$ ...
3
votes
2answers
158 views

Linear Independence of Unipotent Upper Triangular Matrices

Let $A\ne I$ be an upper triangular unipotent matrix in $\mathbb{GL}_n(k)$, where $k$ is a field of characteristic $p$, i.e., $A$ is an upper triangular matrix all of whose diagonal elements are $1$. ...
2
votes
5answers
313 views

How to minimize $\| y- Ax\|$ subject to $\|x\|=1$ and $x \geq 0$?

Given $y \in \mathbb R^n$ and $A \in \mathbb R^{n \times n}$, whis is some way for $$\min_x \| y- Ax\|$$ subject to $\|x\|=1$, and $x \geq 0$ (which means every components of $x$ is nonnegative)? ...
1
vote
0answers
150 views

Transforming matrices in a differential equation

This is from Dupont et al., "Simplified density-matrix model applied to three-well terahertz quantum cascade lasers", PRB 81, 205311 (2010): Equation (3) (...) can be rewritten as a 16x16 system of ...
1
vote
0answers
52 views

Issue with tridiagonal matrix factorization

So let's assume I have an arbitrary mxm tridiagonal matrix made up of real numbers. How many flops are needed to get its QR factorization (assuming I'm using the householder method?) How would one go ...
2
votes
1answer
93 views

When $e$ is an eigenvector to $G$ prove that $e$ is an eigenvector of $G+k I$ and $G^2$

I've just started learning matrices and I've been shown how to perform row operations, how to find an inverse matrix, how to find eigenvectors from a given matrix, reduced-echelon form, basis, dim(M), ...
0
votes
1answer
35 views

Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. $$VV† = -x^2I^2 - y^2J^2 - ...
0
votes
4answers
301 views

Relationships between $\det(A+B)$ and $A+B$

When computing $\det(A+B)$ we notice that there is no relation between $\det A + \det B$. However does the $\det(A+B)$ have any relation to the matrices $A+B$ as they stand?
2
votes
1answer
332 views

$m\times n$ matrix with an even number of 1s in each row and column

So I want to find the number of ways to fill an $m\times n$ matrix with only 0s and 1s such that each row and column has an even number of 1s. I'm pretty stumped here. I've set up m+n equations ...
2
votes
2answers
445 views

How do collinear points on a matrix affect its rank?

Consider the matrix \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{matrix} what effect does $({x_1},{y_1})$,$({x_2},{y_2})$,$({x_3},{y_3})$ being ...
4
votes
3answers
313 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
0
votes
1answer
7k views

Question regarding trivial and non trivial solutions to a matrix.

I would very much appreciate som explanations regarding trivial and non trivial solutions to a matrix (I am a beginner in studies of linear algebra). Suppose that we have two matrices $A$ and $B$. ...
3
votes
5answers
356 views

How do I show that $T$ is invertible?

I'm really stuck on these linear transformations, so I have $T(x_1,x_2)=(-5x_1+9x_2,4x_1-7x_2)$, and I need to show that $T$ is invertible. So would I pretty much just say that this is the matrix: ...
1
vote
1answer
98 views

How to calculate the powers of the following matrix

I need the powers $A^n$ of the following matrix $A=\begin{bmatrix}0 & t & 1-t\\1-t & 0 & t\\ t & 1-t & 0\end{bmatrix}$ with real $t$, but I get messed up in the calculations ...
3
votes
1answer
185 views

The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
3
votes
3answers
226 views

Rotating Matrix by $180$ degrees through another matrix

To rotate a $2\times2$ matrix by $180$ degrees around the center point, I have the following formula: $PAP$ = Rotated Matrix, where $$P =\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$$ $$A= ...
1
vote
1answer
336 views

How to calculate left and right eigenvector corresponding to the zero eigenvalue.

I'm working on $8\times8$ matrix resulting from the Jacobian of $8$ differential equation of a disease model evaluated at disease free equilibrium. I needed to get the left and right eigenvectors ...
11
votes
3answers
222 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
4
votes
3answers
131 views

Order of calculation in all math equations

I already asked a question (Order of operations in rotation matrix notation.) about the order in which a particular equation is "processed" and now I need to generalise that and learn the rules of ...
0
votes
1answer
52 views

$\operatorname{im} A = \ker A$ for a $2 \times 2$ matrix $A$?

Find the image and kernel of $A = \left( \begin{smallmatrix} 2 & 4 \\ -1 & -2 \end{smallmatrix}\right)$. I was told that both the image and the kernel are the lines $y = - \frac{1}{2} x$. Is ...
3
votes
3answers
943 views

Dividing matrices

HI Im learning about matrices in school and am curious about how to divide them so i can find out different identity matrices for refecting shapes over linear functions
1
vote
3answers
213 views

Order of operations in rotation matrix notation.

I'm trying to convert this equation to C# but I'm not a mathematician and I find math notation ambiguous: See the first matrix in this article: http://mathworld.wolfram.com/RotationMatrix.html ...
2
votes
2answers
108 views

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix ...
2
votes
0answers
171 views

Low-rank approximation to the Graph Laplacian matrix of a regular grid.

As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
0
votes
1answer
51 views

If $\operatorname{rank}(A)=k$, can we say something about $\operatorname{rank}(AA^t)$?

Let $A$ a $n\times m$ matrix such that $\operatorname{rank}(A)=k$, can we say something about $\operatorname{rank}(AA^t)$? It's like: If $\operatorname{rank}(A)=m$, can we say anything about ...
0
votes
1answer
66 views

Dimensions of Matrices Range (equalities).

I’d like to find range equalities. Considering the following: $$ A=B+C \\ A=B.C^T \\ A=[ B^T C^T ]^T \\ $$ I would like to find the function $f$ for each equality above. $$ dim( R(A) ) = f( R(B) , ...
0
votes
1answer
2k views

Determine if the matrix is idempotent?

I am dealing with an example to show that the matrix($M = I − X(X'X)^{−1}X'$) is idempotent. X is a matrix with T rows and k columns and I the unit matrix of dimension T. And then to determine the ...
4
votes
1answer
126 views

Condition on trace of product of two matrices to be positive

Given: $A$ is positive definite, $B$ is symmetric and $\operatorname{tr}(B)\geqslant 0$, what could be a minimal additional condition, so that $\operatorname{tr}(AB)\geqslant 0$? ("$B$ is positive ...
1
vote
2answers
121 views

Need help with matrix multiplication: $ (aI + bJ)(cI + dJ) $.

Consider the matrix $$ A = \left[ \matrix{a & -b \\ b & a} \right], $$ and write this as $ A = aI + bJ $, where $$ I = \left[ \matrix{1 & 0 \\ 0 & 1} \right] \quad \text{and} \quad J = ...
3
votes
2answers
59 views

What type of division is possible in 1, 2, 4, and 8 but not the 3rd dimension?

In this article that talks about some history of hamilton http://plus.maths.org/content/curious-quaternions There is a snippet that says this: Multiplication is very sneaky. You can only set up ...
1
vote
1answer
45 views

How does one derive this rotation quaternion formula?

given an angle and an axis, the corresponding quaternion can be computed like this. $w = \cos( Angle/2)$ $x = \text{axis}.x * \sin( Angle/2 )$ $y = \text{axis}.y * \sin( Angle/2 )$ $z = ...
1
vote
0answers
163 views

How is the Quaternion multiplication derived?

Quaternion multiplication seems suspiciously similar to the cross product. How is it derived? Here is a description of the multiplication: Let $Q_1$ and $Q_2$ be two quaternions, which are defined, ...
1
vote
1answer
59 views

Solving for vector in linear algebra

This is not really my field so please bear with me. In the equation: $ (X^TX)\hat{h}_q = X^Ty_q $ I need to find the vector $\hat{h}_q$. $X$ and $y_q$ are known, and $y_q$ is the same length as ...
3
votes
2answers
143 views

If $\operatorname{rank}(A)=m$, can we say anything about $\operatorname{rank}(AA^t)$?

Let $A$ a $n\times n$ matrix such that $\operatorname{rank}(A)=m$, can we say something about $\operatorname{rank}(AA^t)$?
4
votes
2answers
190 views

Math hack for solving system of equations

Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
4
votes
1answer
162 views

What is the difference between first and second right eigenvectors of a row stochastic matrix and their meaning?

In an $n\times n$ non negative row stochastic matrix (rows sum up to 1). The entries of the stochastic matrix I have represent directed links between countries. Why is the first right eigenvector a ...
1
vote
1answer
201 views

Inverse and multiplication of (symmetric p.d.) Toeplitz matrices

Let $A$ and $B$ be two Toeplitz matrices with $A\in \mathbb R^{n\times n}$ symmetric and positive definite, and $B \in \mathbb R^{n\times k}$. I am searching for an elegant proof (or a ...
13
votes
3answers
409 views

If $C$ commutes with certain matrices $A$ and $B$, why is $C$ a scalar multiple of the identity?

I'm self studying Steven Roman's Advanced Linear Algebra, and this is problem 10 of Chapter 8. Let $A,B\in M_2(\mathbb{C})$, $A^2=B^3=I$, $ABA=B^{-1}$, but $A\neq I$ and $B\neq I$. If $C\in ...
0
votes
2answers
205 views

A step in finding the determinant of transpose of a matrix

The following question involves the permutation group, which I am horrible at handling. Any help will be greatly appreciated. Let $A$ be an $n \times n$ matrix with entries $(a_{ij})_{i = 1,2 \cdots, ...
0
votes
2answers
307 views

Matrix Inverse Question

Let $C$ be an invertible 2x2 matrix such that: $$C^{-1} \cdot \begin{bmatrix}1 \\ 2\end{bmatrix} = \begin{bmatrix}3 \\ 4\end{bmatrix}$$ $$C^{-2} \cdot \begin{bmatrix}9 \\ 5\end{bmatrix} = ...
-1
votes
1answer
520 views

How do you determine the matrix which represents T and find range?

Suppose $T:\mathbb{R}^3\to\mathbb{R}^2$ is a linear transformation with: $$T(e_1)=\begin{bmatrix}3\\1\end{bmatrix},\qquad T(e_2)=\begin{bmatrix}4\\1\end{bmatrix},\qquad ...
1
vote
0answers
155 views

How to determine all vectors $b$ for which $Ax = b$ has a solution? Do the columns of $A$ span $\mathbb{R}^3$?

Let $A = \left(\begin{array}{rrr|r} 1&1&-15&36\\ 1&2&-10&41\\ 1&2&-9&42 \end{array}\right)$. Here is the row reduction: $A = \left(\begin{array}{rrr|r} ...
1
vote
1answer
182 views

Prove the matrix is irreducible, stochastic and primitive or not?

The matrix I have below is following which non-negative matrices rules and prove? irreducible? stochastic? primitive? All eigenvalues of the matrix are in a unit circle or not? ...
1
vote
1answer
171 views

Regarding the kernel of a linear transformation and that of the associated representing matrix

Let $V, W$ be finite dimensional vector spaces over a field $F$. Let $\mathcal{B}_{V} = \{\mathbf{v_1, \cdots, v_n} \}$ and $\mathcal{B}_{W} = \{\mathbf{w_1, \cdots, w_m} \}$ be corresponding bases. ...
6
votes
1answer
3k views

Evaluating eigenvalues of a product of two positive definite matrices

Let $A,B\in M_n(\mathbb{R})$ be two symmetric positive definite matrices, i.e.: $$\forall x\in\mathbb{R}^n, x\neq 0, (Ax,x)>0, (Bx,x)>0,$$ where $(\cdot,\cdot)$ is the usual scalar product in ...
2
votes
3answers
1k views

Prove that the rank of a block diagonal matrix equals the sum of the ranks of the matrices that are the main diagonal blocks.

\begin{equation*} X= \begin{pmatrix} A& 0 \newline 0& B \end{pmatrix} \end{equation*} If A and B are some matrices and 0 is a zero matrix, prove that $\ rank(X)=\ rank(A)+\ rank(B)$. Also, if ...
1
vote
1answer
197 views

Book about applied linear algebra to the 3D world?

I have no problem understanding the basics of the linear algebra, but I also know that the linear algebra it's still an analytical approach to real world scenarios that can be solved with matrices and ...
2
votes
1answer
32 views

Bounding the dimension of a subspace associated with a hermitian form.

Suppose that $H$ is an invertible hermitian (self adjoint) matrix in $\mathbb{C}^{n×n}$. Define the hermitian (sesquilinear) form $[x,y]:=x^∗Hy$. Suppose $V$ is a subspace of $\mathbb{C}^{n}$ ...
3
votes
3answers
162 views

Showing that the group of Unitary matrices $(U_n)$ is non-abelian for $n \geq 2$

I know I could show this by counter-example, finding two unitary square matrices of size $2 \times 2$ at least, and conclude $U_n$ is non-abelian. The problem with this, is I think it's somewhat time ...