Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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54 views

What is the right answer to $P^{-1}AP$?

Given the matrices $$A=\pmatrix{b+8c & 2c-2b & 4b-4c \\ 4c-4a & c+8b & 2a-2c \\ 2b-2a & 4a-4b & a+8b \\ }, P=\pmatrix{0 & 1 & 2 \\ 2 & 0 & 1 \\1 & 2 ...
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1answer
46 views

$B=A(I+F)$ with $\|F\|<1$ implies $\|A^{-1}B\|<1/(1-\|B^{-1}A-I\|)$?

Let $A$ be invertible $n\times n$ matrix and $B=A(I+F)$ with $\|F\|<1$ where norm is consistent and submultiplicative. I learned that $I+F$ is invertible and $\|(I+F)^{-1}\|<1/(1-\|F\|)$. So we ...
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1answer
90 views

Inequality involving trace and operator norm

Here's a simple question for which I can't find an answer. Let $W$ be a square real matrix with eigenvalues all real and positive ($W$ is not necessarily symmetric nor diagonalizable) and $A$ a real ...
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2answers
265 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: \[ g(1)=\alpha \] \[ g(2n+j)=3g(n)+\gamma n+\beta_j \] \[ \mbox{for}\ j=0,1\ \mbox{and}\ n\geq1 \] I ...
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2answers
84 views

Is this determinant bounded?

Let $D_n$ be the determinant of the $n-1$ by $n-1$ matrix such that the main diagonal entries are $3,4,5,\cdots,n+1$ and other entries being $1$. i.e. $$D_n= \det \begin{pmatrix} ...
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1answer
65 views

decomposition of m-cycle in m-1 transpositions

I am searching for a proof. Every m-cycle $\sigma = (x_1 x_2 ... x_m)$ can be expressed as an composition of m-1 transpositions. I found many formulas, for example: $\sigma = (x_1 x_2)(x_2 x_3) ... ...
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1answer
95 views

Prove $T ^ {n} = T_ {1} + … + T_ {k}$

Let $ V $ complex inner product space (real) with $\dim V = n$ and let $T $ normal nonzero operator (symmetric) on $V$. Show that there are $k$ operators $T_1, ... T_k: k \le n$ on $V$ such that $T_i ...
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2answers
452 views

irreducible, diagonally dominant matrix

I am facing a problem for irreducible,diagonally dominant matrices. How to prove that irreducible, diagonally dominant matrix is invertible? Please help me in this problem.
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3answers
132 views

Need Help with Proof please help! [closed]

Im not sure really how to do this, will someone please help. Given that $A$ and $B$ are $n\times n$ matrices over field $F$, prove that the solution set $S$ to the matrix equation $XA+BX=0$ is a ...
3
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1answer
90 views

For any matrix norm, is it true $||A|| \le \max|a_{ij}|\cdot ||(1)||$?

Let $|| \cdot ||$ be a matrix norm on $m \times n$ matrices, which is not assumed to be submultiplicative. Is it true that $||A|| \le \max|a_{ij}|\cdot ||(1)||$ where $(1)$ denotes the matrix with all ...
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1answer
182 views

Power series of matrix which is multiplied by a constant factor $c<1$?

(Important: THIS PROBLEM IS NOT DUPLICATED! Note that the case where just one row of $W$ is multiplied by constant $c$, can be handled by the Sherman-Morrison theorem, but the case where the whole ...
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4answers
342 views

The path of complex structure.

$J = \left( {\begin{array}{*{20}{c}} 0&{ - {I_n}}\\{{I_n}}&0\end{array}} \right)$. If $I(t)$ is a path in $\rm M(2n, \mathbb R)$ ($2n \times 2n $ real matrix) such that $I(0)=0$ and ...
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1answer
250 views

Linearly dependent eigenvectors when diagonlising a matrix

If I wanted to diagonalise an $n \times n$ matrix $A$. Let $P$ be the matrix of eigenvectors. Why is it that I need columns of $P$ to be linearly independent? If I had two equal eigenvalues and ...
0
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1answer
43 views

The skew commutativity of two matrices

If $A,B$ are two $n\times n $ real matrices such that $A^2+B^2=-I_n$, then does $AB+BA=0$ necessarily hold?
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2answers
412 views

Kernel of matrices product

Let $A$ and $B$ be two matrices s.t. $\operatorname{Ker}\left(A\right)$, and $\operatorname{Ker}\left( B\right)$ are the null spaces of $A$ and $B$ respectively. What is the ...
4
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1answer
112 views

Name for this simple inequality

Let $x,y$ vectors in $\mathbb{R}^3$. From $$\Vert x+y\Vert^2\geq 0$$ it follows that $$2x\cdot y\geq -\Vert x\Vert^2-\Vert y\Vert^2$$ Has this inequality a name?
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1answer
173 views

Fast way of finding RSS of Multiple Linear Regression

Is there any smarter way to compute Residual Sum of Squares(RSS) in Multiple Linear Regression other then fitting the model -> find coefficients -> find fitted values -> find residuals -> find norm of ...
2
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1answer
70 views

Are these two permutation matrices equivalent?

Definition: The permutation matrix $P_{ij}$ is the identity matrix with rows $i$ and $j$ reversed. When left-multiplied with another matrix, it exchanges rows $i$ and $j$. Am I right in thinking that ...
3
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2answers
60 views

Proper way to “discuss a system”

I am doing exercises to increase my math skills and I found this one: Let $A$ be a real parameter. Discuss the system: $$x - y + 3z = 3A - 1$$ $$2x - (A+1)y -Az = -3$$ I did consider the three ...
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2answers
294 views

Does one always use augmented matrices to solve systems of linear equations?

The homework tag is to express that I am a student with no working knowledge of math. I know how to use elimination to solve systems of linear equations. I set up the matrix, perform row operations ...
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1answer
73 views

Differential Equation: Complex Eigenvalue

For the following system $$ x'=\left( \begin{array}{ccc} \frac{-1}{2} & 1 \\ -1 & \frac{-1}{2} \end{array} \right)x $$ To find a fundamental set of solutions, we assume that $$ x = Ee^{rt}$$ ...
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2answers
66 views

Find all nonzero matrices $A\in M_2(\Bbb Z_3)$ which are not invertible.

Find all nonzero (with every element $\neq$0) matrices $A\in M_2(\Bbb Z_3)$ which are not invertible, and explain why they aren't invertible. It seems simple indeed, but I'm not sure how to solve ...
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3answers
695 views

Find a basis for the subspace $\mathbb{R}^3$ containing vectors

Let $v_1 = \langle 1,0,-1\rangle$ $v_2 = \langle -2,7,2\rangle$ $v_3 = \langle 3,-7,-3\rangle$ I found that these are linearly dependent since I have a free variable upon reducing. However, the ...
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3answers
76 views

If $x = a + b$, and only $x$ is known, how to solve what is $a-b$?

If $x$ equals to $a+b$, how can I solve what is $a-b$, knowing only $x$? (approximation will do as well, if it cannot be solved exactly)
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2answers
655 views

Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
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1answer
42 views

Geometrical meaning of multiplying a non negative matrix

Given two square matrix $X,A \in \mathbb{R}^{N \times N}$ $Y = A^TX$ What is the geometrical meaning of $Y$ if $X$ is non negative? What properties can we claim from $Y$?
4
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2answers
120 views

How to find this greatest common divisor limit?

Let $A^n-I=\begin{bmatrix} a_{n}&b_{n}\\ c_{n}&d_{n} \end{bmatrix} $, where $A=\begin{bmatrix} 3&2\\ 4&3 \end{bmatrix}$, $I=\begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}$ and let ...
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0answers
70 views

Contour Integral of a Definite Positive Matrix

I need to prove the following proposition: Proposition: Let assume that $H\left( \lambda\right) \in\mathbb{R}^{\left( n-1\right) \times\left( n-1\right) }$ is a matrix function such that ...
3
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2answers
541 views

Eigenvalues and Eigenvectors of $X'X$ and $XX'$

I am trying to derive (or prove) the relationship between the eigenvalues and eigenvectors of the matrices $X'X$ and $XX'$. It is fairly intuitive that they are related but I cannot derive the ...
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3answers
113 views

How to find solutions for linear recurrences using eigenvalues

Use eigenvalues to solve the system of linear recurrences $$y_{n+1} = 2y_n + 10z_n\\ z_{n+1} = 2y_n + 3z_n$$ where $y_0 = 0$ and $z_0 = 1$. I have absolutely no idea where to begin. I understand ...
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0answers
34 views

Exceptional Simple Jordan Algebra Cross product

Does anyone happen to know of an explicit construction of the cross product on the exceptional simple Jordan algebra, or perhaps a reference? Context: I'm trying to see if $(D^* a)X b + a X (D^*b) = ...
3
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1answer
574 views

Values for a and b Diagonalizable Over C

For which values of $a$ and $b$ is the matrix $$ \begin{pmatrix} 0 & a\\ b & 0 \end{pmatrix} $$ diagonalizable over $\mathbb{C}$? I know that if $a = -1$ and $b = 1$, then the matrix is ...
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1answer
36 views

Max of two vectors - how is this evaluated?

Suppose that $\max \{\mathbb{v},0\}$ where $\mathbb{v}$ is some vector. When the elements of a vector are all positive or all negative, this is obvious, but what occurs if there are some positive ...
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2answers
76 views

Prove that $ T $ is symmetric

Let $T$ an operator over $V$ given by $$ T(x_{1}, x_{2}) = \left(\displaystyle\frac{x_1}{2}+x_2, \displaystyle\frac{x_1}{4} + \displaystyle\frac{x_2}{2}\right).$$ Prove that $ T $ is an operator ...
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6answers
154 views

Finding a vector $v_3$ such that $\{v_1, v_2, v_3\}$ is a basis of $ℝ^3$ [closed]

Let $v_1 = (2, -1, -2)$ and $v_2 = (1, 2, -2)$ Find a vector $v_3$ such that $\{v_1,v_2, v_3\}$ is a basis of $\mathbb R^3$. Justify your answer. Can someone please help me with this ? I have no ...
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2answers
41 views

$P_2 --> P_3$ Transformation

Let $T: P_2 \to P_3$ be a transformation such that $p(x) \in P_2\mapsto (x+10)p(x) \in P_3 $ 1.Find the image of $p(x) = 5 + x - 3x^2$ 2.Show that $T$ is a linear mapCan anyone show me how to ...
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1answer
154 views

calculate new velocity when the object collides with a vertical wall. [closed]

I'm struggling to solve this question and id be extremely grateful if anyone could lend me a hand. Given the velocity of the object, v, calculate the new velocity, v', when the object collides ...
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2answers
632 views

extending a linearly independent set to a basis

I want to show that every linearly independent set in a finite-dimensional linear space can be extended to a basis for the entire space.
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2answers
56 views

Linear subspace and dimension

If $Y$ is a proper linear subspace of a finite dimensional linear space $X$, I want to show that $Y$ is also finite dimensional and $\dim(Y)< \dim(X)$.
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1answer
49 views

If $T^2 = I$ is a linear operator on $V$, then $V = V^{(-1)}\bigoplus V^{(1)}$.

Let $V$ be a vector space and $T$ be a linear operator on $V$ such that $T^2 = I$. Then $V$ is the direct sum of eigenspaces $V^{(-1)}$ and $V^{(1)}$. I've got that $V^{(-1)}\bigcap V^{(1)} = \{0\}$ ...
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3answers
159 views

Different interpretations of imaginary number

I went through a linear algebra course and I'm a bit confused.. I think I understand the geometric interpretation of imaginary numbers - multiplying by $i$ results in rotation by $90$ degrees in so ...
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1answer
65 views

Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$

Theorem to prove: Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
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2answers
97 views

Revisited: $[T]^{\gamma}_{\beta}$ is diagonal?

Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that ...
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1answer
78 views

Matrix Polynomial Question

Suppose $A$ is a matrix with complex coefficients. Suppose $f(x)$ is a polynomial of minimal positive degree with property that $f(A)=0$. Let $P_A(x)$ be characteristic polynomial of $A$. Prove that ...
4
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1answer
190 views

Solving $Ax = b$ when $A$ is singular

I have a system of equations, expressed as $\mathbf{A} \begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 0 \\ i (\frac{1}{2} + C - a) \\ i(\frac{1}{2} - C - a) \frac{m ...
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0answers
78 views

Does this matrix have any properties?

The matrix is: $\left( \begin{array}{cc} \sin{\theta} & \cos{\theta} \\ \cos{\theta} & \sin{\theta} \\ \end{array} \right) $ I'm interested in its effect on points in the first quadrant, ...
4
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1answer
197 views

Hard-wiring a proof method in my head

There's s a kind of proof regularly used in linear algebra ( proving facts about Transformations, direct sums, basis, ... ) that i have definitely agreed with but still couldn't connect my intuitive ...
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1answer
140 views

How to show by induction that, for $0<\theta<\pi$, $\det A_n=\frac{\sin (n+1)\theta}{\sin \theta}.$

I need help with the underlined part. Thanks in advance Let $A_n$ be the $n\times n$ matrix given by $$a_{ij}= \begin{cases} 0 & \text{if }|i-j|>1, \\ 1 & \text{if }|i-j|=1, ...
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2answers
4k views

Prove that the only eigenvalue of a nilpotent operator is 0?

I need to prove that if $\phi : V \rightarrow V$ is nilpotent, then its only eigenvalue is 0. I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How ...
3
votes
2answers
67 views

Derivative of vectors

I know very little about vector calculus. What is the derivative of $\langle\alpha,\alpha\rangle$ (dot product) and $\alpha^TK\alpha$ and $\langle\alpha,y\rangle$. All these derivatives are by the ...