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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0
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1answer
10 views

Basic linear system with Gauss

I'm studying linear systems and I'm looking for the method to solve this exercize : Find the values of $(a,b,c,d) \ \in \ \mathbb{R}^4 \ \text{such as the linear system with unknow} \ x_1 \ x_2 \ ...
1
vote
1answer
27 views

Find base vectors and dim

Find base vectors and dim of a space described by the following system of equation: $$2x_1-x_2+x_3-x_4=0 \\ x_1+2x_2+x_3+2x_4=0 \\ 3x_1+x_2+2x_3+x_4=0$$ I did rref of the matrix and as a result i get: ...
0
votes
1answer
36 views

Let $A: \mathbb{R}^6 \rightarrow \mathbb{R}^6$ be linear transformation, $A^{26} = I$.

Let $A: \mathbb{R}^6 \rightarrow \mathbb{R}^6$ be linear transformation, $A^{26} = I$. Find linear spaces $V_1, V_2, V_3$, such that: $\mathbb{R}^6 = V_1 \oplus V_2 \oplus V_3$, dim $V_1$ = dim $V_2$ ...
0
votes
2answers
70 views

Why do I get a 1x1 matrix when I multiply a Rx1 vector with a 1xR vector

Today in the lecture the prof wrote something similar to: $$A = \begin{bmatrix} 0 & 0 & 0 &...& 0 & 1 \end{bmatrix}$$ $$B = \begin{bmatrix} b_n & b_{n-1}& b_{n-2} & ...
0
votes
0answers
32 views

Is there a theorem that describes the computation of characteristic polynomial of this matrix?

Suppose I have $$A = \begin{bmatrix} 0 & 1 \\ b & a \end{bmatrix}$$ Then instantly I find that my characteristic polynomial is: $P(\lambda) = \lambda^2 - a\lambda - b$ Expand this matrix ...
12
votes
4answers
1k views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ ...
3
votes
4answers
102 views

Find a linear operator s.t. $A^{2}=A^{3}$ but $A^{2}\neq A$?

From Halmos's Finite-Dimensional Vector Spaces, question 6a section 43, the section after projections. Find a linear transformation A such that $A^{2}(1-A)=0$ but A is not idempotent (I remember A is ...
0
votes
0answers
29 views

Transformation/operation to collapse a series of similar lines into a set of constants?

I have a dataset that can be represented in a matrix form as: $$ Y = \begin{bmatrix} y_{11} & y_{12} & y_{13} & \cdots & y_{1j} \\ y_{21} & y_{22} & y_{23} & \cdots & ...
0
votes
1answer
26 views

Full proof of this matrix propierty

I have to prove the following question: ? I asked yesterday and got answers like "Since $A=Y^{-1}XY, A^2=(Y^{-1}XY)(Y^{-1}XY)=Y^{-1}X^2Y$. So $\alpha A^2+\beta A+γI=Y^{-1}(\alpha X^2+\beta ...
1
vote
1answer
45 views

Queries about square roots of matrices in $M(n,\mathbb R)$

For $A \in M(2, \mathbb R)$ , if $A$ has a square root $B \in M(2, \mathbb R)$ , then $\det A=(\det B)^2$ is non-negative ; and if $s:=\sqrt{\det A}$ ; then using $A^2+s^2I=(trA)A$ , evaluating ...
0
votes
2answers
58 views

Is there a simple way to map this 4x1 vector to this 4x2 matrix?

Is there a simple way to map a vector like $\begin{bmatrix}a\\b\\c\\d\\\end{bmatrix}$ to $\begin{bmatrix}a&0\\b&0\\0&c\\0&d\end{bmatrix}$? I tried to do it via matrix multiplication ...
1
vote
1answer
47 views

Size of conjugacy classes in SL(2,3)

I've been given the representations of the conjugacy classes for a group presentation $G = <x,y,z | x^2 = y^3 = z^3 = xyz>$ which is isomorphic to $SL(2,\mathbb{F}_3)$ which are: ...
1
vote
1answer
47 views

I can't understand one step of this matrix norm proof

I'm reading a multivariable calculus textbook for college, and before a Taylor series proof is given, a lemma is provided with its demonstration. The lemma says: Given a matrix $M(X)= ...
0
votes
1answer
86 views

Show a matrix is invertible [duplicate]

How to show that $$A=\begin{pmatrix}1233&2344&1324&3456\\ 2342&11233&1432&13256\\234132&32432&1234567&43254\\423412&42354&452356&13245\end{pmatrix}$$ ...
2
votes
1answer
75 views

Calculating the inverse components of the Fubini-Study Metric.

In coordinates $(z_1, \dotsc, z_n)$, the Fubini-Study metric, can be written as $ds^2 = \frac{(1 + z_i\overline{z}^i)dz_jd\overline{z}^j - \overline{z}^jz_idz_jd\overline{z}^i}{(1 + ...
0
votes
0answers
32 views

Trace of kernel gram for gaussian kernel

Given a Gaussian kernel $K(h)=\exp(-\frac{h^2}{2\sigma^2})$, the corresponding Gram matrix become $[k]_{ij}=K(x_{i}-x_{j})$. So I think the diagonals of the kernel Gram is unity and hence the trace ...
1
vote
0answers
58 views

How to solve a system of equations with the regular falsi method?

Given a system of equations represented by $Ax = b$ where $A$ and $b$ are known. How can one find vector x using the regular falsi method for a conceptual example where $A$ is 3x3.
2
votes
1answer
104 views

What does an Ulam matrix look like?

I'm trying to visualize an Ulam matrix but I"m having trouble. So it has Aleph one rows and aleph null columns? What do elements of a Ulam matrix look like?
0
votes
0answers
51 views

The eigenvalues after a row and a colum has been deleted from a matrix.

Now I have a zero row sum matrix $L$, and a diagonal matrix $H$, where $L$ can be reviewed as a Laplacian matrix of a directed graph. That is, the off-diagonal elements of $L$ are either $0$ or $-1$, ...
0
votes
1answer
57 views

Vector as a 4x4 matrix

I need to multiply a 4x4 matrix with a 3x1 vector. However, my program only supports 4x4 matrix being multiplied by another 4x4 matrix. How do I represent my 3x1 vector as a 4x4 matrix so the ...
0
votes
1answer
38 views

inverse of matrix n by n

So here is a question that has crossed my mind and never left it: suppose we have a matrix A($n \times n$) and matrix $B(n \times m$), since $(A^{-1})\cdot (B^{-1})$ is equivalent to saying $(A\cdot ...
0
votes
1answer
41 views

What are the elementary divisors of this special matrix

Can someone explain how I can calculate the elementary divisors over $\mathbb{Z}$ (and over $\mathbb{R}$) of the following matrix: $$\begin{pmatrix}2 & -1 \cr -1 &\ddots &\ddots\cr ...
1
vote
2answers
33 views

How can this be proven (Matrices)

I need to prove why the image on the bottom is true, btw this is on a matrices unit so you know that the order of multiplication does matter
-1
votes
2answers
12k views

Matlab function, rotation matrix

Is this the correct way to calculate a rotation matrix for a given angle around a unit vector, i am having problems verifying it. ...
6
votes
1answer
229 views

Does similarity of integer matrices with square $-I$ imply the transition matrix is an integer matrix?

I'm working on a homework question, and I'm stuck. The question is: Let $A$ and $B$ be $2n \times 2n$ rational matrices with $A^2=B^2=-I$. The first part of the question asks to show that $A$ ...
0
votes
1answer
72 views

Solving simultaneous equations of matrices

so the question which is written in swedish states the following, given that all the varibles in the simoultanous equation are MATRICES of size 2*2 and B is given to be (2 1 1 1) (idk how to insert ...
1
vote
3answers
86 views

Projection onto subspaces - point to line projection

In the following document about projection onto subspaces, the author is computing the transformation matrix to project a vector $b$ onto a line formed by vector $a$. Since the projected vector ...
4
votes
1answer
134 views

Closest Matrix with Specific Eigenvector

Consider a vector ${\bf x}$ and a matrix $A_0$ with $A_0(i,j)\ge0$. What is the best way of getting matrix $A$ s.t. $$A = \arg \min |A-A_0|$$ subject to $$A{\bf x} = \lambda {\bf x} \hspace{2mm} ...
5
votes
3answers
263 views

If $(A-\lambda{I})$ is $\lambda$-equivalent to $(B-\lambda{I})$ then $A$ is similar to $B$

When reading the topic about primary and rational canonical form of matrices I stuck myself on this theorem: The matrices $A,B\in K^{n\times n}$ are similar if and only if their characteristic ...
1
vote
1answer
41 views

Apply Cayley transformation on vector x

If I have $Q = (I + S)(I - S)^{-1}$ ($Q$ is the Cayley transformation of skew-symmetric matrix $S$) then how do I construct a rank-2 $S$ such that $Qx$ has all zeros except the first component?
1
vote
1answer
80 views

Solving matrix AX=B matrix for X

I'm having the following matrices in the form A X= B.Where A & B matrix are known values. How can i solve for the matrix X?. Please provide me some formulas or Ideas for solving this. Note: we ...
0
votes
1answer
62 views

When PSD, singular value is equal to eigenvalue

It is known that If a matrix is PSD (symmetric), then its eigenvalues are equal to its singular value. How to prove it? Hope for a hint. thanks,
1
vote
1answer
20 views

property of simultaneous diagonalisable matrices

If A,B and B,C and C,A are simultaneously diagonalisable matrices then is this true that A,B,C are simultaneously diagonalisable? Here $A,B,C\geq{0}$.
0
votes
1answer
38 views

How to prove tr($Z^TZ$) is the sum of singular values

Ask a maybe trivious question: We know the product of singular values is |determinant of tha matrix| as in: Singular value proofs Then how to prove: $tr(Z^TZ)^{1/2}$ $=\sigma_1(Z) +...+ ...
0
votes
7answers
4k views

Proof: The inverse of the inverse matrix is the matrix.

If $A$ is a square matrix such that it is not singular, then $(A^{-1})^{-1} = A$ How can I prove this property? I would appreciate it if somebody can help me.
1
vote
1answer
45 views

Bounding a Symmetric Matrix

Consider the following $n \times n$ matrix $A$, which has 1's on the superdiagonal and subdiagonal and 0's elsewhere, i.e. $$\begin{pmatrix} 0 & 1 & 0 & \cdots & \cdots & \cdots ...
1
vote
2answers
96 views

Orthogonal Matrix question

$$A= \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} $$ is an orthogonal matrix. a) Prove that $A^{-1}=A^T$ b) show further that $a^2=d^2$ and that $b^2=c^2$. ...
-2
votes
2answers
5k views

Proof: The identity matrix is invertible and the inverse of the identity is the identity

How can i show that: $II^{-1} = I = I^{-1}I$ (the identity matrix is invertible) for all cases. And then proof that: $I^{-1} = I$ (The inverse of the identity is the identity). I don't know how start ...
1
vote
1answer
90 views

Cayley transformation of a skew-symmetric matrix is orthogonal?

If $S$ is skew-symmetric ($S^{T} = -S$), how do I show that $Q$ is orthogonal where $$Q = (I + S)(I - S)^{-1}$$ which is the Cayley transformation of $S$.
1
vote
1answer
49 views

Find the Least Integer $k$ such that $B^k=I$

If $A$ and $B$ are two non Singular Matrices such that $B\ne I$, $A^6=I$ and $$AB^2=BA$$ Then what is the Least Integer $k$ such that $B^k=I$ My Try: Given $$AB^2=BA$$ which we can write as ...
3
votes
2answers
30 views

$f:M(n,\mathbb R) \to \mathbb R$ be , then $\exists ! C \in M(n,\mathbb R)$ such that $f(A)=Trace (AC) , \forall A \in M(n,\mathbb R)$?

Let $f:M(n,\mathbb R) \to \mathbb R$ be a linear function , then does there exist a unique $C \in M(n,\mathbb R)$ such that $f(A)=Trace (AC) , \forall A \in M(n,\mathbb R)$ ?
0
votes
0answers
10 views

Estimation of changes in solution x when A change

Suppose I have system $Ax = b$ where A = [${2}$ $-1$ $1$; $-1$ $10^{-10}$ $10^{-10}$; $1$ $10^{-10}$ $10^{-10}$]; b = [$2(1 + 10^{-10})$; $-10^{-10}$; $-10^{-10}$] and x = [$10^{-10}$; $-1$; $1$] ...
0
votes
2answers
68 views

Unable to calculate pseudo-inverse $A^TA$

I'm trying to calculate the pseudo-inverse of $A^TA$ as described in this paper: The SVD is particularly simple to calculate when the matrix is of the form $A^TA$ because $U=V$ and the rows of ...
0
votes
1answer
19 views

Storing a real matrix - fl notation

I have two matrix $A$ and $B$ and $fl(A), fl(B)$ denote the stored version them, respectively. Let $fl(AB)$ be the stored version of the product of $A$ and $B$. Is it true that $fl(AB) = fl(A) * ...
37
votes
6answers
42k views

Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric? I seem to remember a proof similar to this from my linear algebra class, ...
1
vote
0answers
34 views

Minimize $f(X)=trace\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}\right)$

Minimize $$f(X)=trace{\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}X\right)}$$ subject to the condition $g(X)=det(X)=1$. Then for taking $X=\begin{bmatrix} ...
4
votes
1answer
708 views

Transpose a square matrix code

I know it's not programming area , but I think it's more related to math. I have the following function: ...
2
votes
2answers
74 views

Linear Algebra: determine the number of linearly independent columns

$$ A = \left[\begin{array}{cccc}0& 1& 1\\ 1& 2& 3\\ 2& 0& 2\end{array}\right]$$ Clearly first and second columns are linearly independent. The third column is the sum of ...
0
votes
2answers
70 views

Linear Algebra Matrices

Determining the values of a for which the Matrix A has an inverse ! A= \begin{pmatrix} 1 & a & 1 \\ 2 & a+2 & 1 \\ 1 & 2 & a \end{pmatrix} How i solved it: ...
0
votes
1answer
97 views

Schatten p norm p>1

The Schatten p norm is differentiable away from the origin for p> 1. Does a stronger condition of Lipschitz continuity of the gradient also hold?