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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30 views

$rank(T^n) = rank(T^m)$ for any positive integer $m \geq n$

Let $T$ be a linear operator on a finite dimensional space $V$. Prove that if $rank(T^n) = rank(T^{n+1})$ for some positive integer $n$, then $rank(T^n) = rank(T^m)$ for all positive integer $m \geq ...
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2answers
43 views

Prove the matrix $ \left( \begin{array}{ccc} B & A^T \\ A & 0 \\ \end{array} \right)\ $ is nonsingular [closed]

Suppose the matrix $A\in\mathbb{R}^{m\times n}$, $m\leq n$, and has full row rank $m$, $B\in\mathbb{R}^{n\times n}$ is a symmetric, $Z\in\mathbb{R}^{n\times(n-m)}$ is the matrix whose columns span ...
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1answer
35 views

Differentiation map and the Cayley-Hamilton theorem

I have computed (a) to be $-\lambda^3$. I also know that the Cayley-Hamilton theorem states that substituting the matrix A (where A is matrix with p(λ)=det(λI-A) for λ in this polynomial results in ...
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3answers
80 views

Solve for $\textbf{x}(t)$ from the system of differential equations $\textbf{x}' (t) = A \textbf{x}(t)$.

Solve for $\textbf{x}(t)$ from the system of differential equations $\textbf{x}' (t) = A \textbf{x}(t)$, where $$ A =\begin{bmatrix} 0 & -1 & 0\\1 & 0 & 0\\ 0 & 0 & ...
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1answer
40 views

If A is a Hermitian matrix then SAS* is Hermitian

If $A$ is an $n\times n$ Hermitian matrix, and $S$ is an nxn matrix, then $SAS^*$ is also Hermitian. Why is this true? I have seen this claim made in several places but can't find a proof.
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2answers
86 views

Is the Square Root of an Inverse Matrix Equal to the Inverse of the Square Root Matrix?

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
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1answer
43 views

What are these tick marks after the x, y, and z called?

What are these marks called and what do they stand for? This is for a Affine Transformation.
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35 views

How to calculate row sums of a power of a matrix

Let $P $ be an $n\times n$ matrix whose row sums $=1$.Then how to calculate the row sums of $P^m$ where $m $ is a positive integer?
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3answers
49 views

How do I prove matrix irreversibility without determinants?

I have to prove that if matrix has two identical rows or columns then it is not a reversible matrix. I know that in such scenario matrix determinant is equal zero, but I cannot use determinants in my ...
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3answers
39 views

Eigenvectors of $\left( \begin{array}{ccc} 0 & -b \\ a & 0 \end{array} \right)$

This is similar to my previous question in that I when I form a system of simultaneous equations and solve them all the terms cancel and I don't get any information on the eigenvectors. The matrix in ...
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3answers
44 views

Determinant of linear transformation

Given a linear transformation $T:V\rightarrow V$ on a finite-dimensional vector space $V$, we define its determinant as $\det([T]_{\mathcal{B}})$, where $[T]_{\mathcal{B}}$ is the (square) matrix ...
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1answer
50 views

Is there a way to do this besides brute force?

$A$ is a $d\times n$ matrix and $\mu>0$. I'm trying to show that $$(AA^T + \mu I)^{-1} A = A(A^T A+\mu I)^{-1}.$$ The only way I've thought about doing this was by the brute force method of ...
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1answer
52 views

Calculate the Determinant of a NXN matrix

Is there any elegant way to calculate the determinant of the N X N symmetric matrix M, where the $(i,j)$ term is defined by: $$M_{ij}=m_i+m_j$$ with $0\le m_i, m_j \le1$ The solution will be in ...
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2answers
73 views

Matrix Algebraic Operations, If AA = AB, does A = B?

A and B are 2 x 2 matrices and A is not a zero matrix. How is the following proof incorrect? Since AA = AB, AA - AB = 0 A (A - B) = 0 and since A does not equal zero, then A - B = 0, therefore A = ...
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1answer
30 views

rank of a submatrix

Suppose the $8 \times 4$ matrix $A$ has rank $4$. Is it always true that any $4 \times 4$ submatrix of $A$ has rank $4$? I am doing research on coding theory and I am wondering whether this is true. ...
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1answer
28 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$(A)$ ?
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1answer
62 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 ...
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2answers
47 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 ...
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2answers
30 views

General Solution Of Linear Equations

$x_1+x_2-6x_3+4x_4=6$ $3x_1-x_2-6x_3-4x_4=2$ $2x_1+3x_2+9x_3+2x_4=6$ I have row reduced the matrix and got $$\left(\begin{array}{cccc|c} 1 & 1 & -6 & 4 &6\\ 0 & 1 & -3 ...
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3answers
61 views

Span of columns (or rows) of a given matrix?

Consider the following matrix: $$A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \end{pmatrix}$$ The columns of $A$ span $\mathbb{R}^2$. The columns of $A$ span $\mathbb{R}^3$ ...
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1answer
55 views

Under what conditions are the eigenvalues of a matrix finite?

Suppose we have a square matrix $A$. Under what conditions on $A$ ensure that all eigenvalues of $A$ are finite?
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1answer
40 views

Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is: Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} ...
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1answer
126 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
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1answer
42 views

How to factor and reduce a huge determinant to simpler form? Linear Algebra

So, I have learned about cofactor expansion. But the cofactor expansion I know doesn't reduce the number of rows and colums to one matrix. I usually pick a colum, multiply each element in the column ...
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1answer
33 views

Express summation in terms of matrix norm

Express the following $$\sum _{ i=1 }^{ n }{ ({ \beta }_{ 1 }x_{ i }+{ \beta }_{ 0 }-y_{ i })^{ 2 } }$$ To become something of the form: $∥Ax−b∥^{ 2 }$ where $A$ is an $m$−by−$n$ matrix and $b$ is ...
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2answers
93 views

On the nilpotence of the matrix $AB-BA$ [closed]

Given $n\times n$ matrices $A,B$ satisfy: $rank(AB-BA)=1$ Prove that $(AB-BA)^{2}=0$ Generalize the problem if possible. Any solution not mention Jordan canonical form would be appreciated!
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1answer
44 views

Solving an augmented coefficient matrix so there are infinitely solutions

I am trying to figure out this math problem. For what values $a,b$ does the linear system have infinitely many solutions? This is the matrix $$ \left[ \begin{array}{ccc|c} ...
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3answers
58 views

Tridiagonal Symmetric Matrix

Could anyone help me to find the determinant of a $N\times N$ tri-diagonal symmetric matrix, named "$A[i,j]$" with $i,j \le N$, that has all the elements in the super-diagonal and sub-diagonal equal, ...
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1answer
42 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
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2answers
57 views

Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
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2answers
111 views

Matrix-Multiplication

I have to matrices: $$A=\pmatrix{1&a&1\\1&0&a\\1&2&0} ; \quad B= \pmatrix{1&b&3\\2&1&0}$$ The task is to determine $AB, AB^T, BA$ I think i cannot ...
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1answer
37 views

Matrices as linear transformations

I am reading a proof which claims: A matrix of $m\times n$ is a linear transformation from $m$ vector-space to $n$ vector-space, And therefore, by the dimension theorem: $m = \dim\ker A + ...
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2answers
54 views

Find a nonzero matrix $A$ in $M_{2\times 2}(R)$ satisfying $v\cdot Av=0$ for every $v\in R^2$

Ok, so I have this problem: Find a nonzero matrix $A$ in $M_{2\times 2}(R)$ satisfying $v\cdot Av=0$ for every $v \in R^2$. So if I say that $v=\begin{bmatrix}x\\y\end{bmatrix}$ and that ...
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2answers
41 views

Let L be a linear transformation defined by its standard matrix AL

$\begin{bmatrix}1&-1&1&2\\2&-2&3&4\end{bmatrix}$ As I understand domain and codomain of L are $L:\mathbb{R}^4 \rightarrow \mathbb{R}^2$ ? How can I write formula definition ...
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1answer
64 views

Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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3answers
51 views

How to check whether this matrix is diagonalizable or not.

Let $\rm A$ be a complex $3\times3$ matrix with $\rm A^3=-1$. Which of the following statements are correct: $\rm A$ has three distinct eigenvalues. $\rm A$ is diagonalizable over $\Bbb ...
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1answer
61 views

Solve for A from A x transpose(A)

I'm sure that I knew how to do this once many moons ago and that it's really simple. I have a matrix X which is defined as: $$ X = AA^T $$ How do I find A? (I know X) Thanks! Fiona
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3answers
46 views

examining if a matrix is diagonizable

I was practicing some linear algebra problems and I stopped at this one: Without calculating the eigenvectors, show that the following matrix is diagonalizable and find the diagonal matrix to which ...
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2answers
134 views

Trace minimization of a matrix

Suppose $S = \pmatrix{1&1\\ 1&0\\ 0&1}$, $W$ is a $3\times3$ covariance matrix, which could be regarded as fixed. I need to find a $2\times 3$ matrix $Q$ that minimizes $$ ...
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1answer
46 views

Can someone look at my proof about the convergence of $e^{-tA}$

Hi I am trying to prove that if A is a symmetric positive definite matrix then $e^{-tA}\rightarrow 0$ as $t\rightarrow\infty$. So I have attempted an answer but I'm not sure it is correct. ...
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2answers
76 views

Set of linear equations

Find eigenvalues and eigenvectors of the matrix: $\begin{pmatrix} 1 & 0 & -2 \\ 1 & 3 & -1 \\ -1 & 0 & 2 \end{pmatrix}$ $\begin{pmatrix} 1-\lambda & 0 & -2 \\ 1 & ...
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2answers
44 views

Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
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1answer
130 views

Linear Algebra matrix notation

My question is referring to the following $4 \times 6$ matrix: $$\begin{bmatrix} 0 & 1 & 2 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 ...
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2answers
67 views

Eigenvalues of $A+B$ in this special case

Let $A$ and $B$ are real, square matrices with the same dimension. We know that $\text{rank } A = 1$ and we know the eigenvalues of $A$. Furthermore, we know that $B$ has only zeros in the diagonal, ...
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2answers
46 views

Prove that if $B=P^{-1}AP$, then $q(B)=P^{-1}q(A)P$

Is it possible to prove this using a similarity invariant? For example showing that $$\det(q(B))=\det(q(A))$$
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1answer
45 views

Find the last column of a matrix. Find the matrix.

$A\left[\begin{matrix} 1 & 0 \\ 0 & 0 \\ 1 & 1 \end{matrix} \right]$ = $\left[\begin{matrix} 2 & 3 \\ -1 & 0 \\ 5 & -7 \\ 0 & 6 \end{matrix} \right]$ (1)Find the last ...
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2answers
54 views

b such that Ax = b has no solution having found column space

$A:=\begin{bmatrix} 2 & 6 & 0 \\ 3 & 1 & 3 \\ 1 & 0 & 0 \\ 4 & 8 & 1 \end{bmatrix}$ I've found the basis for the column space by doing row reduction (i.e. ...
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1answer
54 views

Possible typo in math Linear Algebra textbook

In the section of my Linear Algebra textbook, it states the coefficient matrix $\lambda$I - A can be written in this form \begin{pmatrix} \lambda-a_{11}&a_{12}&... ...
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1answer
33 views

Inverse Identity + Constant Matrix

I need to invert a square symmetric matrix $$ C = c\, I+cs\, B $$ Where: (1) $B$ is a constant matrix of 1 for each entry. (2) $c$ and cs are just positive real numbers. (3) $I$ is the identity. ...
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3answers
65 views

Is the set of diagonalizable, complex matrices open in the set of square matrices?

Is the set of complex, diagonalizable matrices open in the set of square matrices? I asked myself this question and I tried to prove it somehow. However, I don't have any good approach so far. ...