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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7 views

Number of ways two matrices can be multiplied?

Given the dimensions of two matrices what are the different ways they can be multiplied? Example A[2][2] and B[2][2] then answer is 2. Let the dimensions of first matrix be n x m and second be m x p
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0answers
16 views

Prove that the determinant of a given matrix is proportional to the area of the triangle whose corners are the three points.

For three points in 2D, $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the determinant of \begin{bmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 2\\ x_3 & y_3 & 3\\ ...
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2answers
23 views

45 degree rotation of the line y=-3x+1?

Currently working on problems in a textbook for Senior Maths, however I'm currently at a problem where my answer, despite attempting it multiple times, is incorrect to the textbook's result. The ...
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2answers
16 views

Find initial system from solution of a RREF matrix.

I was looking at the solution's of my exam. I used a different technique: I first found the RREF of the matrix giving the solution. Then I worked backward to find the initial system. Can anyone ...
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0answers
8 views

Residual Sum of Squares

Through transformation with orthogonal matrix O, the problem b(hat)=arg minb||y−Xb||2 is equivalent to b(hat)=arg min||y∗−X∗b||2, where y and y∗ are in Rm, X and X∗ are in Rm×n (m≥n) and y∗=Oy and ...
2
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0answers
15 views

Consistent Augmented Matrix

Well, the linear system which at least has one solution is called "consistent" linear system. Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system: ...
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3answers
123 views

Is $A$ invertible if $ABAB^2 = I$?

Q: Have two matrices of order $5$, $A$ and $B$. If $ABAB^2 = I$, is $A$ invertible? A: Yes, the inverse of $A$ would be $BAB^2$ My definition of an inverse matrix is: For some matrix $X$, ...
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3answers
29 views

Why does a matrix column being a multiple of another imply that the matrix is singular?

Q: If a matrix of order $9$ has a column that is a multiple of another, does the system $Ax = 0$ has infinite solutions? A: Yes, because the matrix would not be invertible. Can you ...
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2answers
57 views

Linear independence of matrices $I, A, A^2$

I want to prove that $I,A,A^2\:$matrices $\in M_{2\times 2}$ are $\textit {linearly independent}$. I consider the following matrices and their "corresponding" vectors: $I=\begin{pmatrix} 1 & 0 ...
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1answer
14 views

Rotating a plane defined by a normal and a distance from the origin around an arbitrary point in 3D space

I have a plane defined by its normal and its distance from the origin. I have a rotation matrix and a point in 3D space around which to do the rotation. What formula will allow me to do the rotation? ...
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27 views

Cross product uniqueness

I have following relationship between vectors $A_1'(t)=\psi(t)\times A_1(t) \tag1$ $A_2'(t)=\psi(t)\times A_2(t) \tag2$ $A_3'(t)=\psi(t)\times A_3(t) \tag3$ Given Data " ' " means derivative ...
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19 views

Proof for a rank-one Decreasing step

I came across this result in a paper in my area concerning rank-one decompositions. However, I am unable to understand one of the steps. I am reproducing the result here. Let $X$ be a $N\times N$ ...
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13 views

Second order derivative,coordinate transformation,Jacobian

Any one help me please. If (x,y)=f1 (u,v) and N=f2 (x,y) Then {∂N/∂x; ∂N/∂y}=J-1 {∂N/∂u); ∂N/∂v} Here J=[∂x/∂u ∂y/∂u ∂x/∂v ∂N/∂v]; Similarly I need to ...
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2answers
22 views

Show that $\pm 1$ are the only eigenvalues of the linear operator $f$ as the transpose of a matrix

Let $V$ be the vector space of $n\times n$ matrices over $K$ under addition and let the linear operator $f$ be given by $f(A)=A^{T}$, where $A^T$ denotes the transpose of matrix $A$. Show ...
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1answer
58 views

Matrix multiplication: is $\begin{bmatrix} 3 \\ 2 \\ 1 \\ \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ \end{bmatrix}$ defined?

Is this matrix multiplication is possible? Microsoft mathematics gives me an answer for it? How it could be a correct? If we need multiple two matrix, number of rows and columns should be equal. So ...
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1answer
16 views

Matrix Multiplication in an Inequality

Assume I have the following equation: $$ A_1 \vec x_1 \circ \vec y_1 \gt A_2 \vec x_2 \circ \vec y_2 $$ Here $ A_1 $ and $ A_2 $ are some n x n matrices and $ \vec x_1 , \vec x_2 , \vec y_1 , \vec ...
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2answers
22 views

Rank One decrease

Let $X$ be a $N\times N$ real positive semi-definite(p.s.d) matrix with rank $R$. Let $x_1\in Range(X)$ be a non-zero $N\times 1$ vector such that $X_1=X-x_1x_1^T$ is still p.s.d. What is the rank of ...
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0answers
29 views

Prove residual sum of square through transformation with orthogonal matrix

Through transformation with orthogonal matrix O, the problem b(hat)=arg minb||y−Xb||2 is equivalent to b(hat)=arg min||y∗−X∗b||2, where y and y∗ are in Rm, X and X∗ are in Rm×n (m≥n) and y∗=Oy and ...
-1
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1answer
26 views

Linear Algebra Orthogonal Real Matrix [on hold]

Let O be an nxn orthogonal real matrix, i.e. OTO=In, where In is the identity matrix. Prove that 1) Any entry in O is between -1 and 1. 2)If gamma is an eigenvalue of O, then absolute value of gamma ...
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1answer
32 views

Equation involving a partial trace

Is there, in general, a solution to the following equation? $\text{Tr}_{V_1}(A(X\otimes I_{V_2})) = B$ where A is an operator on $V_1\otimes V_2$, $B$ is an operator on $V_2$, $I_{V_2}$ is the ...
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1answer
30 views

How do you diagonalize this matrix and find P and D such that A = PDP^-1?

1 1 4 0 -4 0 -5 -1 -8 I3 = 3x3 identity matrix λ 0 0 λI3 = 0 λ 0 0 0 λ λ-1 -1 -4 = 0 λ+4 0 5 1 ...
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1answer
46 views

Inequality of Positive-definite matrix.

In this question matrix $A$ is positive-definite if and only if $\forall x\ne0 :x^TAx>0$. ($A$ is not necessarily symmetric) Let $D$ be a positive-definite matrix such that it has block form: ...
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4answers
54 views

Rotating Frame and Angular Velocity

We have an equation $ \frac{dr}{dt}=\Omega \times \bf r \tag 1$ SPECIFICATIONS $\times$ means cross product,$\Omega$ constant angular velocity,${\bf r}$ is the postion vector of an object Given ...
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0answers
26 views

Formal inverse of a matrix ressembling Fourier's matrix

What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$? When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and ...
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0answers
6 views

Limitations of Dominance matrices [on hold]

Does anyone know any Limitations of Dominance Matrices when ranking sports teams?
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2answers
52 views

Proof that $A^2 = A$ where A and B are square matrices , if $BA = B$ and $AB = A$. What did I do wrong?

The problem Given that for square matrices $A$ and $B$ of the same order, $AB = A$ $BA = B$ Prove that $A^{2} = A$. My proof $$ \text{Starting with the given condition ,}\\ BA = B\\ ...
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0answers
25 views

Inverse of matrix with index notation

While reading quantum field theory I've come across a matrix in index notation that has to be inverted. For example the matrix $$\eta_{\rho\sigma}p^2-p_\rho p_\sigma+m^2\eta_{\rho\sigma}-i\epsilon ...
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30 views

Shrinkage operator for matrices

Here http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf, on page 188, you can see the derivation of the soft thresholding operator or shrinkage operator for the case of vectors using Moreau ...
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0answers
32 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
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2answers
39 views

matrix powers problem

let $ A $ be the matrix :\begin{bmatrix}1 & 3 & 1\\4 & 2 & 3\\2 & 1 & 1 \end{bmatrix} Prove that $A$ verifies the expression : $ -A^{3}+4A^{2}+12A+5 I_{3} = O_{3}$ Deduct ...
3
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1answer
38 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
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0answers
14 views

CDF of smallest eigenvalue of non-central Wishart matrix - how to evaluate the integral.

Does anybody know how to derive the distribution of the smallest root of a non-central Wishart matrix? I have got an integral expression that would give me the desired answer but cannot solve the ...
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1answer
24 views

Finding the Formula of the Product of $e_{i,j}$ and $e_{k,l}$ to Return Zero Matrix

My teacher for calculus this year gave a handout on the first day with an excerpt from Rings, Fields, and Vector Spaces by B.A. Sethuraman. The reason for this is in the beginning of Sethuraman's book ...
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1answer
42 views

Deriving equation in vector notation

I had some trouble deriving an equation from the book 'Elements of statistical Learning' p. 108 equation 4.9. This heavily relies on linear algebra, so I was wondering how the author came to his final ...
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2answers
37 views

Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
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2answers
95 views

Binomial Theorem on a Matrix

Does the expression follow binomial theorem? $(A + I)^n$ where $A$ is matrix, $I$ is identity matrix. I know the binomial theorem but do not know whether it is applicable to matrices also.
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11 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
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1answer
41 views

Permutation matrices

Let $\mathscr{M}$ be the set of all $n\times n$ matrices having entries $0$ and $1$ in such a way that there is one $1$ in each row and column. (a) If $M\in\mathscr{M}$, describe $AM$ in terms of ...
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0answers
42 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
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16 views

Generalized Schur complement theorem

Let $M$ be an $(n+m)\times(n+m)$ real non-symmetric positive semidefinite (PSD) matrix partitioned as \begin{eqnarray*} M=\left(% \begin{array}{cc} A~~B\\ C~~D\\ \end{array}% \right), ...
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1answer
20 views

The nullity of a square matrix with linearly dependent rows is at least one. TRUE OR FALSE

Here is the answer my textbook gives. http://imgur.com/ycCRoWK I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
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1answer
35 views

TRUE OR FALSE: Matrices with linearly independent row and column vectors are square.

Here is the answer of my textbook: http://imgur.com/vEoY31O Why must a matrice with linearly independent vectors have nullity(A)=0? That is where I lose track of the question. Are zero rows ...
2
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3answers
77 views

$A^2=cA$ for some $c \neq 0$

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: $$A^2 = cA.$$ Questions. 1) Is there a matrix class for matrices ...
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0answers
16 views

Leading eigenvalues of large sparse unsymmetric matrix

I have a matrix R which is sparse and all eigenvalues are -ve with a zero eigenvalue. Size of R is more than 1 million X 1 million. But I need to calculate only few large (by value not by magnitude) ...
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1answer
25 views

Prove that a normal matrix is unitary/Hermitian

I'm stuck with these two questions for while. I'd appreciate your help. ...
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0answers
16 views

Exponential and power of a special bidiagonal matrix

Given the bidiagonal matrix $$ \mathbf{A}=\begin{bmatrix} a_1 & b_1 & 0 & 0 & \dots & 0 & 0\\ 0 & a_2 & b_2 & 0 & \dots & 0 & 0 \\ 0 & 0 ...
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22 views

Immannt of a matrix.

I want to know in details about immanant of matrix. I have come to know about it in here but it could not give me sufficient knowledge. Kindly provide me some link where I can get introductory ...
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1answer
39 views

Normal Matrix Having all real eigen values is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
3
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1answer
53 views

Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
5
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2answers
106 views

An interesting linear algebra question

Let $A$ and $u$ be $n\times n$ matrix and $n\times 1$ vector of $\mathbb{C}$. Denote $\overline{A}$ is the matrix $(\overline{A})_{ij}=A_{ij}^*$, the conjugate number; ($\overline{A}$ is not the ...