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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0answers
60 views

Reducing a linear algebra expression to quadratic form

I am trying to solve the following exercise for my Machine Learning course. Expand this expression so that there are only quadratic terms: $(\mathbf{x} - \mathbf{\mu})^T \mathbf{\Sigma}^{-1} ...
1
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5answers
50 views

Matrices to the power of $n$ and their reversibility

Please forgive my ignorance. I am busy with a first year course in elementary linear algebra and there are some concepts I do not grasp. Particularly, questions regarding matrix invertibility. For ...
3
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2answers
48 views

example of complex structure with negative determinant

is it possible to find a matrix $J_1 \in GL(4,\mathbb R)$ such that $\det J_1=-1 $ and $J_1^2=-\operatorname{id}$ ? if it is, how can we prove that every matrix $M \in GL(4,\mathbb R)$ such that ...
0
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0answers
14 views

Simplify recursion function based on a matrix, real-world usecase

I have an auction running, and I'm trying to calculate the expected amount of first, second etc. places to be taken by a particular bid. To achieve that, based on historical data I make a following ...
5
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3answers
257 views

How to encode matrices uniquely

Given a square matrix $A=[a_{ij}]_{n \times n}$, an operation $swap(A, i, j)$ is defined to swap row $i$ and $j$ of $A$ and do the same thing with the corresponding columns. For example, in the ...
0
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2answers
48 views

Flipping a matrix?

Real quick question: I was wondering, how would one denote mathemathically the flipping of a matrix, horizontally or vertically, around its own axis?
0
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1answer
21 views

Transformation of a surface normal

I'm taking a university level course in discrete geometrics and graphical programming, and I'm having trouble understanding this exercise. Let p be a point in R^3, n a surface normal, and M a ...
3
votes
1answer
35 views

What should be the characteristic polynomial for $A^{-1}$ and adj$A$ if the characteristic polynomial of $A$ be given?

Let the characteristic polynomial of $A$ be $\psi_A(x):=p(x)$. If $A$ be non-singular, then find that the characteristic polynomial of $A^{-1}$ and adj$(A)$. My attempt: We have \begin{align*} ...
4
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1answer
79 views

Why are matrices written as such?

Another thread has talked about the purpose of a matrix. Dr. Math roughly summarized it as: A matrix is just a compact notation, which allows you to specify several linear equations at once ...
0
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1answer
32 views

Solve $3$ variables using $4$ equations where $1$ equation contains $3$ variables

Suppose we are given the system of equations $$\alpha_1A+\beta_1B+\gamma_1C=x$$ $$\alpha_2A+\beta_2B+\gamma_2C+\theta_2D=y$$ $$\alpha_3A+\beta_3B+\gamma_3C+\theta_3D=z$$ where ...
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1answer
23 views

properties of largest eignvalue of product of two matrices

I'm searching for the proof of this lemma it's about largest eignvalue of product of two matrices. one of them is positive definete and the other one is symmetric. B is symmetric matrix, A is Positive ...
0
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2answers
43 views

Is there a multiplication transformation that will add the bottom row of a matrix to the top row?

Given matrix $$ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $$ Is there a matrix $B$ such that: $$ AB = \begin{bmatrix} a+g & b+h & c+i ...
3
votes
1answer
13 views

Prove that $A$ is invertible when $a_0 \not=0 $ and $A^{-1}=q(A)$ for some polynomial $q$.

Let $p(\lambda)= (-\lambda)^n + a_{n-1}\lambda^{n-1} + ... + a_0$ be characteristic polynomial of matrix $A$. Prove that $A$ is invertible when $a_0 \not=0 $ and $A^{-1}=q(A)$ for some polynomial $q$. ...
0
votes
1answer
17 views

Infinite-dimensional version of Gram matrix is invertible

We all know that a Gram matrix (a matrix with entries that are inner products of basis functions) is a invertible. Suppose I have $a_{ij} = (h_i, h_j)_H$ where the $h_j$ are basis functions of a ...
0
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1answer
24 views

Cholesky Decomposition and Orthogonalization

I recently came across a methodology for orthogonalizing variables that are collinear, that uses Cholesky Decomposition, but I am not entirely grasping the intuition of it. Let' assume we have three ...
1
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1answer
17 views

Matrices admit a QR decomposition

I just wanted to ask which matrices admit a QR decomposition. I think that all matrices $A \in \mathbb{R}^{m \times n}$ with $m \ge n$ admit a QR decomp. Are these the only ones that have a QR decomp, ...
0
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0answers
15 views

help me find the gimbal locks

I have this transformation (x, y, z) |-> (x'', y'', z''). How can the gimbal locks be discerned and where are they? ...
1
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1answer
39 views

What is the relationship between three points on a quadratic curve and the curves coefficients?

In other words, is there a formula to get the coefficients a,b and c in terms of three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3, y_3)$? I am asking this because I have a linear algebra problem that ...
1
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1answer
25 views

How to find trace of adj$A$ from the characteristic polynomial of $A$?

Let the characteristic polynomial for $A$ be $t^n+c_1 t^{n-1}+c_2t^{n-2}+\cdots+c_{n-1}t+c_n$. From it, is it possible to find the trace of adj$(A)$ ?
4
votes
3answers
104 views

How to get determinant of $A$ in terms of tr$(A^k)$?

Suppose that $A$ is $n$-square matrix such that $t_r:=$ tr$(A^r), r=1, 2, \cdots, n$ are given real numbers. How shall we compute $\det(A)$ in terms of $t_r$s? I am completely unable to do this. ...
3
votes
1answer
38 views

Condition for linear minimal polynomials

I'm just wondering that there is a necessary and sufficient condition for minimal polynomials for in which cases are them linear. Let $A$ be a square matrix. I think that $A$ has a linear minimal ...
0
votes
1answer
18 views

What is the sample variance-covariance matrix?

This is a more succinct question from a previous post, but I have arrived at two different answers, and need help determining which - if either - is correct. I start with a 4*3 matrix: ...
1
vote
2answers
110 views

How do I show that $1$ is not an eigenvalue for $A$, by showing that there are no eigenvectors for $\lambda = 1$.

Consider the matrix $$A=\begin{pmatrix}-1 & 3& 3& 3\\ 3& 1& -1& 5\\ 3& -1& 7& -1\\ 3&5& -1&1\end{pmatrix}.$$ How do I show that $1$ is not an ...
3
votes
1answer
74 views

a linear algebra problem arising in geometry

This is a matrix problem. Assume that $A$ and $B$ are real $n\times n$ matrices. Denote $\Lambda=A+iB$, $$ M=\left (\begin{array}{cc} A &-B\\ B & A \end{array} \right ) $$ I would like to ...
1
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1answer
38 views

Rotation Matrix to Quaternion(proper Orientation)

Given Data in the figure In this figure we have a unit vectors $ x,y,z$ as axis. Axis of rotation is $b$ and angle of rotation is $\phi$. $\phi$ is unknown and $b$ is given as $b= \frac{1}{2 ...
1
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1answer
21 views

How to do reduce rows with Wolfram Alpha over certain set

If I want to row reduce a matrix: $$ \begin{matrix} 1 & -1 & 0 & 4 \\ 2 & -2 & 1 & 3 \\ 5 & -5 & 1 & 15 \\ \end{matrix} $$ ...
1
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0answers
15 views

Prove one of the eigenvector entries has the smallest magnitude

Let $L\in \mathbb{R}^{n \times n}$ be the Laplacian matrix of a simple undirected graph and $D_i$ be the same size matrix with $i$th diagonal element $1$. Denote the smallest eigenvalue of $L+D_i$ as ...
1
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2answers
21 views

Show that $p(A) = \begin{bmatrix} p(A_{11})&(Mess)\\ 0&p(A_{22})\\ \end{bmatrix}$ for any polynomial $p(x)$. (See problem for full question.)

Important Note: This is a homework problem. The full question is as follows: If $$A = \begin{bmatrix} A_{11}&A_{12}\\ 0&A_{22}\\ \end{bmatrix}$$ where $A_{11}$ and $A_{22}$ are square, ...
1
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2answers
30 views

Rank of $I_m - X_{m \times m}$ given rank of $X$

I have a matrix $X_{m\times m}$ which is idempotent and has $rank(X) = n < m$. I have for some time now been trying to calculate $rank(I_m - X)$ but have been unable to do so. I should be able to ...
0
votes
1answer
18 views

When are two 3D Lines parallel in Plücker matrix form?

When are two lines in 3 dimensional space parallel, when the lines are both represented by Plücker matrices $L$ and $L'$. I'm trying to prove the solution to this question: ...
0
votes
0answers
16 views

Checking hand manipulations of matrices

Beginning with a 4*3 matrix: 5 4 -1 2 3 -3 3 4 -4 1 3 -2 I have to perform four manipulations on it, which I did by hand. I wanted to ask if my thinking and/or ...
0
votes
0answers
27 views

Find the limit of this matrix as its power approaches infinity

Find the matrix power, Ak, of A = (v1,v2) v1 = (p,1-p) v2 = (1-p',p') Where v1 and v2 are column vectors, and 0 <= p <= 1, 0 <= q <= 1, p /= q. ...
0
votes
0answers
30 views

Quaternion Integration - Initial value problem

We have a standard form of quaternion integration equation $$ q(t) = q(t_0) \exp\left(\frac 12 \int_{t_0}^t \mathbf{\omega}(\tau) d\tau\right),\tag 1 $$ For reference you can check equation (42) in ...
0
votes
0answers
28 views

Quaternion Equivalence

Assume $R_{3\times3}$ is a rotation matrix. Question Is it true that there exists two quaternions representing this same rotation matrix $R_{3\times3}$ ? Hint : $\theta = \arccos\left( ...
-2
votes
2answers
59 views

$A^2-7A-6I=0$, where $I$ is the identity matrix. Show that matrix $A$ is invertible and find $A^{-1}$. [closed]

Given a matrix $A$ and suppose $A^2-7A-6I=0$, where $I$ is the identity matrix. Show that matrix $A$ is invertible and find $A^{-1}$.
0
votes
0answers
39 views

Determinant - derivation of the general formula and its history [duplicate]

I know the formula for calculating matrix determinant. What's I'm wondering is where did that general formula come from? And why determinants are so important? Obviously they are useful in finding ...
0
votes
1answer
17 views

If $p(x)=x^2-cx$ annihilates $A$, then $A$ is similar to $c \operatorname{diag}(1,\dots,1,0,\dots,0)$.

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $A^2 =cA$. I had a question about this matrices, and I get an anwser, ...
5
votes
1answer
70 views

Conic section: What is the coordinate matrix of its bilinear form?

Given is the conic section $x^2 + xy + y^2 + 2x +3y - 3 = 0$. I need to find the coordinate matrix $M_\beta(s)$ of the bilinear form $s: \mathbb{R}^2 \times \mathbb{R}^2 -> \mathbb{R}$. I read ...
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votes
0answers
28 views

Why does $ax=b$ consistent imply that a invertible, if A is nonzero

Here I have a question: Why does $Ax=b$ consistent imply that $A$ is invertible,if $A$ is nonzero What I know: $Ax=b$ consistent implies there is at least one solution. If $A$ is invertible, then ...
0
votes
2answers
38 views

A quick clarification about elementary row operations?

I was solving a homework question that asks me why 3R2 - 2R1 is not an elementary row operation. Is it because we are doing subtraction instead of addition? However, I feel like that it not it. I mean ...
2
votes
2answers
33 views

$A \in Gl(n,K)$ if and only if $A$ is a product of elementary matrices.

I want to show the statement: $A \in GL(n,K)$ if and only if $A$ is a product of elementary matrices. I could show the reverse implication: Suppose $A=T_1...T_m$ where each $T_k$ is an elementary ...
2
votes
3answers
64 views

To find the determinant of a matrix

Given $A_{n\times n}$=$(a_{ij}),$ n $\ge$ 3, where $a_{ij}$ = $b_{i}^{2}$-$b_{j}^2$ ,$i,j = 1,2,...,n$ for some distinct real numbers $b_{1},b_{2},...,b_{n}$. I have to find the determinant of A. I ...
1
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1answer
21 views

Finding that values k that make this matrix invertible without using the determinant

The matrix in question is A = [(1,1,1),(1,2,k),(1,4,k^2)]. I know that I can row reduce the matrix to rref, which should in theory leave me with some k values in the matrix from which I can see what ...
8
votes
3answers
198 views

Quick way to find eigenvalues of anti-diagonal matrix

If $A \in M_n(\mathbb{R})$ is an anti-diagonal $n \times n$ matrix, is there a quick way to find its eigenvalues in a way similar to finding the eigenvalues of a diagonal matrix? The standard way for ...
3
votes
1answer
32 views

Find inverse of $I+\mathbf{ab}^\intercal$

Could you guys give me some hints on this homework? Find inverse of $\mathbf{I} + \mathbf{ab}^\intercal$. Hint: try to form $c\mathbf{I} + d\mathbf{ab}^\intercal$ and solve for $c,d$. What happens ...
0
votes
1answer
28 views

How to find system of equations from solution space

I have to find homogeneous system of linear equations whose solution space is: V = span((1,-2,4,3),(1,-1,6,4),(3,-8,8,3)). First I found vectors were linearly dependent, so I discarded the third ...
0
votes
1answer
17 views

Determine an orthonormal basis so that $s(v_i, v_j) = 0, 1 \leq i, j \leq 3, i \not= j$

Determine an orthonormal basis $ (v_1, v_2, v_3) $ so that $ s(v_i, v_j) = 0, 1 \leq i, j \leq 3, i \not= j $ $s$ is a symmetrical bilinear form given by the matrix A: $$ A = M_\beta(s) = ...
1
vote
0answers
50 views

Decomposing a stochastic matrix into a product of stochastic matrices.

It is well-known that any square real matrix of small rank $k$ can be decomposed into a product of a skinny matrix with $k$ columns and a fat matrix with $k$ rows by means of an SVD. This question is ...
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votes
1answer
20 views

Calculating the signature of matrix A?

given is a symmetrical bilinearform s that has the following matrix: $A = M_\beta(s) = \begin{pmatrix} -3&0&-1\\0&-3&0\\-1&0&-1\end{pmatrix}$ I have to calculate the ...
-1
votes
2answers
52 views

If a matrix is non diagonalizable, what other method can I use to calculate the nth power?

First off, I have this matrix A: 1 0 3 1 0 2 0 5 0 I have calculated the eigenvalues, which are ...