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
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
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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
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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
73 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
36 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 ...
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1answer
23 views

All possible echelon forms of $3 \times 2$ nonzero matrix [closed]

Well, I'm looking for all possible echelon forms of $3 \times 2$ nonzero matrix. Is there any way to generalize the it to find all possible echelon forms of $m \times n$ matrix? Thanks
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: $$ \begin{matrix} 1 & -1 & 0 & 4 \\ 2 & -2 & 1 & 3 \\ 5 & -5 & 1 & 15 \\ \end{matrix} $$ ...
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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 ...
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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
29 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
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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
25 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
24 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
27 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
36 views

Determinant - derivation of the general formula and its history

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, ...
0
votes
2answers
41 views

Let $A$ be an $n\times n$ real matrix and $V = \operatorname{span}\{I, A, A^2, \dots\}$. Show that $\dim(V) \leq n$. [closed]

Let $A$ be an $n\times n$ real matrix and $V$ be the vector space over $\mathbb{R}$ spanned by $\{I, A, A^2, \dots\}$, where $I$ is the $n\times n$ identity matrix. How can I prove that $\dim(V) \leq ...
2
votes
1answer
51 views
+100

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
35 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
63 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
19 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
194 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
31 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
21 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
16 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
48 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
50 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 ...
0
votes
1answer
32 views

How to solve matrix eigenvalue equation which has a summation.

General problem: If I have some $n \times n$ matrices $\mathsf{M}^\tau$, and column vectors (with $n$ rows) $X^\tau$ is there some mathematical tricks I can do to solve the eigenvalue equation $ ...
1
vote
1answer
27 views

What is the significance of a matrix squared

I have a question as follows: The stylised map below shows the bus routes in a holiday area. Lines represent equivalent routes that run each way between the resorts. Arrows indicate one-way scenic ...
2
votes
0answers
29 views

To find the volume dilation, integrate the determinant of the Jacobian

On the road toward proving the change of variables theorem in several variables, is there a painless way to show that $$\text{Vol}(\phi(U))=\int_{U}|\text{det}(d\phi)|,$$ where $\phi$ is $C^1$, ...
2
votes
1answer
32 views

Do lines between determinants pass through the inverse?

Let A be a $2 \times 2$ matrix whose inverse also exists. If I was to draw a line from each of the 3 vertices (that are not the origin) of the determinant of A, to the 3 vertices of the determinant of ...
1
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2answers
59 views

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then ...
0
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1answer
24 views

finding number of submatrices of a matrix of given order?

How do we calculate the number of possible submatrices of a matrix of order $5\times 6$? options for the answer are: $465$ $1953$ $2048$ $30$
1
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2answers
30 views

Determining the values of $b$ in $Ax=b$ to have a consistent system.

$$\begin{bmatrix} 1 & -4 & 7\\ 0 & 3 & -5\\ -2 & 5 & -9 \end{bmatrix}\begin{bmatrix}g \\ h \\ k\end{bmatrix}$$ What are the real values of $g,h,k$ for the system to be ...
0
votes
1answer
37 views

What is a good reference for learning about induced norms?

Wikipedia tells me a little about it. Following the wiki-link treasure hunt leads me to topics such as "p-norms on finite dimensional vector spaces". Which makes me want to ask: what's a good ...
1
vote
1answer
46 views

Common eigenvectors implies commutativity

I am stuck on a seemingly simple problem: if $\mathbf{M},\mathbf{N}$ are $n\times n$ and have all eigenvectors in common, then $\mathbf{MN}=\mathbf{NM}$. I can prove this if they are diagonalisable, ...
0
votes
1answer
33 views

Nullspace and Base of 2 by 2 Matrix [closed]

Hope someone can help me to solve and understand the following problem : $ f:M_{22} (\mathbb{R}) \rightarrow M_{22} (\mathbb{R}) $ $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ $$f(A) ...
0
votes
1answer
28 views

Column space of a matrix product

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times q}$, $C\in\mathbb{R}^{n\times q}$ be matrices such that $$C=AB$$ where $C$ and $B$ are full column rank. Then do $C$ and $B$ have the same ...
0
votes
3answers
49 views

Determinant question $\det(A^{-1/2}) = \det(A)^{-1/2}$

Can someone show me how: $\det(A^{-1/2}) = \det(A)^{-1/2}$ where we assume that $A$ is invertible. thanks
0
votes
1answer
33 views

Implications of Positive Definiteness

Assuming I have a complex non-symmetric matrix $A$ which is "positive definite" in the sense that $\Re(x^*Ax) > 0$. A necessary and sufficient condition for $A$ to be "positive definite" is that ...
1
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0answers
23 views

Calculate rotation on a sphere with given coordinates

I have a sphere with a fixed radius. I have a set of points on that sphere, let's say $p_1, p_2$ and $p_3$ and it's $3$D Cartesian coordinates. I rotated each of the points around the center of the ...
0
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0answers
73 views

What does a '$-$' mean in front of a matrix?

I feel ridiculous for asking this, but I can't seem to find a clear answer. Let $$U = \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$$ Show that $U$, $-U$, $-I$ where $I$ is the $2 \times 2$ ...
1
vote
2answers
39 views

Calculus on Matrices [closed]

I have a basic doubt regarding calculus involving matrices. Dimensions of each matrices are also indicated along matrix name Question If I have a matrix $\kappa(s)_{3\times 1}$ what is ...