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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2answers
32 views

Determine the rank of the linear map given by a $n \times n$ matrix , dependend on n. Proof by induction

The task: Let $$ A:= \begin{pmatrix} 1 & a & a & ... & a\\ a & 1 & a & ... &a \\ a & a & ... & a & a\\ ... & ... &... & 1 & a \\ a ...
0
votes
2answers
31 views

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible.

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible. I'm having trouble proving this, I tried saying that $|AB|=|A||B|=0 \implies |A|=0 \text{ or } |B|=0$ but that got ...
2
votes
1answer
10k views

What's the relationship between singular, nontrivial and linear dependent?

I understand that if a matrix is singular, it has no inverse. If it has nontrivial solutions, it means at least one solutions exists. If it is linearly dependent, it means that for $a_1 ...
3
votes
1answer
185 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A,B \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ A ...
8
votes
2answers
60 views

Interesting Array of Integers with Strange Pattern

I was experimenting and I found this pattern: Start with an (infinite) array with top row with all ones, and leftmost two columns also all ones. $$ \begin{matrix} 1 & 1 & 1 ...
1
vote
0answers
70 views

Why covariance constraint subsumes the average power constraint?

I am studying an optimization problem in the form of \begin{equation} \begin{aligned} &\underset{p(x)}{\text{maximize}} & & W\\ & \text{subject to} & & 0 \preceq K_{X} ...
5
votes
2answers
263 views

Is there any inverse-commutator for matrices

My question is very simple. Given a symmetric real matrix $A$, and a square real matrix $C$, how can one solve the equation $[A,X]=C$, where $[A,B]$ is commutator of $A$ and $B$, i.e., $[A,B]=AB-BA$. ...
5
votes
5answers
106 views

Is it possible to have an $n\times n$ real matrix $A$ such that $A^TA$ has an eigenvalue of $-1$?

Question: Is it possible to have an $n\times n$ real matrix $A$ such that $A^TA$ has an eigenvalue of $-1$? I can prove that it is not possible for $n=1,2$, but I am not sure for the general ...
0
votes
4answers
215 views

Why if the columns of a matrix are not linearly independent the matrix is not invertible?

Why if the columns of a matrix are not linearly independent the matrix is not invertible? I have watched this video about eigenvalues and eigenvectors by Sal from Khan Academy, where he says that for ...
0
votes
0answers
26 views

Matrix multiplication in a different representation

What I would like to do is figure out how to get the expression; $$\begin{bmatrix} x_{1} & x_2 \\ \end{bmatrix} \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} ...
0
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1answer
22 views

The computation of rotation matrix

I got stuck on this quick computation which involve a rotation matrix. Suppose $R\in \mathbb R^{N\times N}$ is a rotation matrix, i.e., $|R|=1$ and $RR^T=I$. Let us write $R=(R_{ij})$, $1\leq i,j\leq ...
3
votes
0answers
97 views

Find transformation matrix with respect to another basis

I understand how we can find the transformation matrix $D$ with respect to another basis $B$, by using a transformation matrix that we already know, say $A$: $$D = C^{-1}\cdot A\cdot C$$ Where $C$ is ...
0
votes
0answers
15 views

Stone Representation Theorem on Matrix Rings?

I would like to know whether Stone Representation Theorem http://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras has some usages or applications on matrix rings. More ...
5
votes
1answer
150 views

Finding the Jordan Canonical form of a $6 \times 6$ matrix

Find the Jordan Canonical Form of the following matrix $$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 ...
4
votes
2answers
479 views

Help in finding the Jordan canonical form of a matrix

Determine the Jordan Canonical Form of the following matrix: $$A=\begin{bmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4\\ \end{bmatrix}$$ I am trying to determine the Jordan ...
0
votes
0answers
22 views

Eigendecomposition; find eigenvectors

This seems like a fairly easy problem but I don't know how to solve it. Consider two matrices: $\mathbf{A}$ (3x3) and $\mathbf{V}$ (3x3). I know that the two are related by: $$\mathbf{A} = ...
2
votes
2answers
68 views

Lower bounding the eigenvalue of a matrix

Suppose I have the following symmetric matrix $$ A'=\begin{bmatrix} A + b b^T & b \\ b^T & 1 \end{bmatrix} $$ where $A$ is positive definite $n \times n$ symmetric matrix and $b$ is a $n ...
0
votes
1answer
3k views

Common coefficient matrix of two matrices

Consider the following two systems. (a) \begin{array}{ccc} 4 x - 2 y &=& -3 \\ x+ 5 y &=& 1 \end{array} (b) \begin{array}{ccc} 4 x - 2 y &=& 2 \\ x+ 5 y &=& 3 ...
3
votes
1answer
50 views

Identity for natural log of matrices

Let $A$ and $B$ be square matrices such that $\ln (A)$ , $\ln (B)$ and $\ln (AB)$ are all defined. Is it true that $$\ln (AB)= \ln (A) + \ln(B)$$ only if $AB=BA$ ? I appreciate any help, Thanks
1
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0answers
48 views

Inverse of $R^T D R$ where $R$: rectangular and $D$: diagonal

Is there any formula for the following triple product: $$(R^T D R)^{-1}$$ where $R$ is rectangular and $D$ is diagonal? The real situation is like this. I have the equation $$Ax = b$$ which is a ...
0
votes
1answer
39 views

How do I find such matrices $X_{1},\ldots,X_{9} \in \mathrm{M}_{2}(\mathbb{Z}) $?

Is there someone who can give at a least an idea for solving this problem? Determine the matrices $ X_{1} , X_{2} , ..., X_{9} \in \mathrm{M}_{2}(\mathbb{Z})$ such that: $$(X_{1})^{4} + ...
0
votes
1answer
58 views

Continuity of the eigenvalues.

I came across a statement of which I do not understand the meaning. The smallest eigenvalue of a $k \times k$ symmetric matrix $M$, $\inf_{ \{v \in R^k | ||v|| = 1 \} } v'Mv$, is continuous in ...
3
votes
2answers
71 views

Inverse of the Toeplitz matrix

I am working on the inverse of the sum of an identity matrix and a Toepltz matrix, and trying to find the formula for the (1,1) element of the inverse. For example, Assume $c$ is a nonzero constant, ...
2
votes
1answer
67 views

Quotient group and adjoint matrix

The exercise 1211 in "Problems and Solutions in Mathematics" by Ta-Tsien: Let $M$ be an $n \times n$ matrix of integers. Suppose that $M$ is invertible when viewed as a matrix of rational numbers. ...
0
votes
1answer
43 views

Dimension of the image and the kernel of f

I've got the linear map: f: $\mathbb{R^4} \rightarrow \mathbb{R^3}$ with $ \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right) \mapsto \left( \begin{array}{c} x_1+2x_2+x_3 \\ ...
1
vote
1answer
212 views

A question about the positive definite matrices and condition number

Let $A=LDL^T$ be a symmetric positive definite matrix, where $L$ is a unit lower triangular matrix, and $D=\textrm{diag}(d_{ii}).$ Show that $$\textrm{Cond}_2(A) \geq \frac{\max (d_{ii})}{\min ...
0
votes
0answers
52 views

How to find the location of a point in a global coordinate system from a local coordinate system

I was wondering if you would be able to help guide me on a solution involving rotation matrices. In-terms of data, I have the global coordinate system a $3\times 3$ matrix, the local coordinate ...
3
votes
1answer
30 views

Convergence of powers of products with diagonal matrices

Suppose that $M$ is an $n\times n$ matrix with $\rho(M)<1$ (i.e. its maximum absolute eigenvalue is less than $1$). Is the following statement then true? If $\forall t\in\mathbb{N}$, $D_t$ is ...
1
vote
0answers
53 views

Characterization of $(0 ,1)$-matrices under swap of columns or rows

I have a squared $(0,1)$ matrix. How I can know if two $(0, 1)$ matrices are the same under column and rows swapping? Moreover, how I can characterize the equivalence classes of $(0,1)$ matrices ...
0
votes
0answers
346 views

Prove if A and B are skew symmetric then A+B is skew symmetric

If $A^T$ = $-A$ which means A is skew symmetric then prove that $(A+B)$ is also skew symmetric. I managed to prove it like this: $(A+B)^T$ = $A^T$+$B^T$ =$(-A+-B)$=$-(A+B)$ Therefore ...
0
votes
1answer
49 views

Linear combination to recover particular data entry from denoised data?

Let $\mathbf{x} = [x_1, x_2, x_3]^t$ the 'data' where $x_1$ is considered to be 'noise', $M$ a $3\times 3$-matrix with full rank, and $\mathbf{y} = M\mathbf{x}$ the obserced mixture. Let $m^-_i$ ...
1
vote
1answer
39 views

how to understand this identity about the range in linear algebra

I see the identity in page 48 of paper http://arxiv.org/pdf/0909.4061.pdf. Specially, if $U^TU=UU^T=I$, then we will have $U^T\text{range}(M)=\text{range}(U^TM)$, where $\text{range}(M)$ means the ...
1
vote
0answers
25 views

Prove that matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
1
vote
1answer
27 views

Prove or disprove the following statement (matrix equation)

Let $A,B \in \Bbb{C}^{n \times n}, B$ invertible and $$ 5A^2+3A-5B^T(B^{-1})^T=0 $$ Then $A$ is invertible. What I did: $(B^{-1})^T=(B^T)^{-1}$ so: $5A^2+3A-5B^T(B^{-1})^T=0 \iff 5A^2+3A-5I=0$ I ...
-2
votes
2answers
80 views

Let $\left| {{a_{ii}}} \right| > \sum\limits_{i \ne j} {\left| {{a_{ij}}} \right|} $.Why does $A$ is nonsingular? . [duplicate]

Let $A \in {M_n}$ and $\left| {{a_{ii}}} \right| > \sum\limits_{j \ne i} {\left| {{a_{ij}}} \right|} $.Why does $A$ is nonsingular?
4
votes
2answers
5k views

Strictly diagonally dominant matrices are non singular

I try to find a good proof for invertibility of strictly diagonally dominant matrices (defined by $|m_{ii}|>\sum_{j\ne i}|m_{ij}|$). There is a proof of this in this paper but I'm wondering ...
0
votes
1answer
323 views

how to check if matrix is separable or not? [closed]

how could I know that a matrix is separable or not? for example, consider the following matrix: Is it separable onto sums or even products?? How to check that??
2
votes
0answers
30 views

Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
1
vote
0answers
100 views

How to find the conjugate of a matrix

To find the adjoint of a matrix first we have to find the conjugate of matrix. for a 3X3matrix \begin{bmatrix} 1&-1& 1 \\ 1&2 & 2\\1&1&2 \end{bmatrix} some one explain me how ...
0
votes
1answer
29 views

Why the following $dim(N(A)) + dim(C(A^T)) = n$ is true?

I was studying about row, column, null and left null spaces, and one thing I don't understand is why the dimension of the null space of a matrix $+$ the dimension of a the row space is equal to the ...
2
votes
4answers
123 views

How to compute determinant of $n$ dimensional matrix?

I have this example: $$\left|\begin{matrix} -1 & 2 & 2 & \cdots & 2\\ 2 & -1 & 2 & \cdots & 2\\ \vdots & \vdots & \ddots & \ddots & \vdots\\ 2 & 2 ...
6
votes
1answer
1k views

Conditions for a real matrix to have real eigenvalues

The eigenvalues of a symmetric real matrix are all real. I was wondering if there are conditions either more general than symmetry or that may or may not overlap with symmetry to ensure eigenvalues to ...
-2
votes
1answer
69 views

A question in matrix norm. [duplicate]

Let $A \in {M_n}$ and $\left\| {\left| . \right|} \right\|$ be a matrix norm on ${M_n}$.Why does ${\left\| {\left| A \right|} \right\|_2} \le \left\| {\left| A \right|} \right\|_1^{\frac{1}{2}}\left\| ...
1
vote
2answers
122 views

How to determine the transition matrices when doing Gaussian elimination?

Guassian elimnation can be done by swapping or adding rows, and by multiplying rows by scalars, etc. We use it to bring for example our original matrix to an upper triangular matrix, so that we can ...
3
votes
2answers
89 views

Eigenvalues of a block diagonal matrix

The matrix $A$ below is a block diagonal matrix. Each block $A_i$ is a 4 $\times$4 matrix with known eigenvalues. $$A= \begin{pmatrix}A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots ...
6
votes
1answer
190 views

Evalute big determinant

Today in exam I tried to evaluate this determinant but failed, only somehow "guessed" the answer I got here. Now in home I've managed to find something intuitive, just want to know whether the ...
0
votes
0answers
53 views

How to find a real valued similarity transform for block diagonalization

I have a real-valued square matrix $A$ with $n$ eigenvalues with zero real part and $m$ eigenvalues with non-zero real part. How do I find a real-valued similarity transform $T$ such that $A = ...
1
vote
0answers
36 views

Solving System of Linear Equations

These are the two known equations: $$\frac{(I_2+I_3)-(I_1+I_4)}{I_1+I_2+I_3+I_4} = \frac{2x}{L}$$ $$\frac{(I_2+I_4)-(I_1+I_3)}{I_1+I_2+I_3+I_4} = \frac{2y}{L}$$ where I know the values of $(x,y,L)$. ...
0
votes
1answer
44 views

Consider the linear transformation $L:R^n\to R^n$ defined by $L\left(X\right)=AX$, then $A$ is diagonalizable iff the matrix of $L$ is diagonal.

I was asked to study the following corollary I could only understand up to theorem 3, does anyone know what the name of this corollary is and if there is clearer proof online?
2
votes
4answers
116 views

The trace identity $\text{tr}((A+B)^2) = \text{tr}(A^2) + \text{tr}(B^2) + 2\text{tr}(AB)$

Prove that $$\text{tr}((A+B)^2) = \text{tr}(A^2) + \text{tr}(B^2) + 2\text{tr}(AB).$$ Else show a counterexample. I've tried using the trace properties such as $$\text{tr}(A+B) = \text{tr}(A) + ...