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
votes
5answers
243 views

Find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$

We want to find an expression for $A^n = \left( \begin{array}{cc} 1 & 4 \\ 2 & 3 \end{array} \right)^n$ for an arbitrary "n". I have tried writing out a few elements of the sequence as $n \to ...
1
vote
2answers
64 views

Eigenvalues of a transition probability matrix

I have read that, for $$I - \alpha P$$ where $I$ is the $n\times n$ identity matrix, $\alpha \in (0,1]$, and $P$ is the transition probability matrix with dimensions $n \times n$, $I - \alpha P$ is an ...
1
vote
1answer
36 views

Is $\left\|A^TA(x-y)\right\| = \left\|A^TA\right\|\times \left\|x-y\right\|$ correct? $A \in \mathbb{R}^{n \times n}$

In the derivation of following, I meet a dumb problem: Note: 1. $\left\|\: \cdot \,\right\|$ is the $l_2$ norm. 2. $A \in \mathbb{R}^{n \times n} $ 3. $x,y \in \mathbb{R}^{n}$ ...
0
votes
0answers
101 views

eigenvalues of $A^TA$ and $AA$

I am a little bit confused about such fundamental problems: Suppose 1. $Ax=\lambda x$. 2. $A \in \mathbb{R^{n \times n}} $. Case I: $$A^TAx = \lambda A^Tx=\lambda \lambda x=\lambda^2x$$ ...
1
vote
1answer
43 views

Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
0
votes
2answers
179 views

QR decomposition proof

Let $A\in\mathbb{M}_{m\times n}(\mathbb{R})$ with $m>n$ and $rank(A)=n$ and take the decomposition $A=QR$ with $Q\in\mathbb{M}_{m\times n}(\mathbb{R})$ a orthogonal matrix and ...
2
votes
1answer
52 views

Is my algorithm correct? (Polar decomposition)

I cant seem to find my mistake. Consider this matrix $T = $\begin{bmatrix} 2 & 1 & 1 \\[0.3em] -1 & 2 & 0 \\[0.3em] 0 & 1 & -1 \end{bmatrix} I need ...
0
votes
1answer
74 views

Upperbound on the following logarithmic function with matrix

I am trying to find an upperbound the expression below with a function $f$ that is a function of the identity matrix $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...
1
vote
2answers
142 views

Finding orthogonal projections onto $1$ (co)-dimensional subspaces of $\mathbb R^n$

1)Consider the vector space $\mathbb{R}^n$ with usual inner product. And let S the subspace generated by $u\in \mathbb{R}^n,u\neq 0$. Find the orthogonal projection matrix $P$ onto the subspace ...
1
vote
1answer
157 views

How to find that two adjacency matrices are equal

What is the easiest way to tell if these two graphs are isomorphic and how do I know which nodes in both graphs are the same. I've made the adjacency matrices but they are pretty big. I think I need ...
2
votes
1answer
90 views

Proof: $|||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| $?

I am looking for a proof of the following: \begin{equation*} |||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| \end{equation*} Where A, B are positive, hermitian matrices, and $|||⋅|||$ ...
3
votes
3answers
84 views

Why does $\frac{1}{{\left\| {\left| {{A^{ - 1}}} \right|} \right\|}} \le \left\| {\left| B \right|} \right\|$?

Let $A,B \in {M_n}$ suppose that the following statements are true: $A$ is nonsingular, $A+B$ is singular, $\left\| {\left| . \right|} \right\|$ is matrix norm. Why is it true that: ...
2
votes
2answers
75 views

Algebric and geometric multiplicity and the way it affects the matrix

Given a matrix $A$. Suppose $A$ has $\lambda_1,\dots,\lambda_n$ eigenvalues each with $g_i$ geometric multiplicity and $r_1,\dots,r_n$ algebric multiplicity, $g_i\leq r_i$. Given this information ...
4
votes
1answer
102 views

What operations can I do to simplify calculations of determinant?

My question is simple. Given an $n \times n$ matrix $A$, what operations can we do to the rows and columns of $A$ to make the calculation of its determinant easier? I know we can put it into row ...
0
votes
2answers
49 views

I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $$\begin{align} T\left(-2 x^2\right) &= 3 x^2 + 3 x \\ T(0.5 x + 4) &= -2 x^2 - 2 x - 3 \\ T\left(2 x^2 - 1\right) ...
1
vote
2answers
114 views

Rotation matrix

I'm finding different results for the 3D rotation matrix in the XY plane from different sources and I was hoping for someone to help clarify. In my "applications of vector calculus" book, the matrix ...
0
votes
2answers
87 views

methods of constructing a matrix from its null space span

I have a matrix of size $4\times3$ and its null-space span is $\{(1,2,3), (2,5,7)\}$. How can I find the original matrix? It is not obvious from the span which vectors are free.
10
votes
4answers
5k views

Symmetric matrix is always diagonalizable?

I'm reading my linear algebra textbook and there are two sentences that make me confused. (1) Symmetric matrix $A$ can be factored into $A=Q\lambda Q^{T}$ where $Q$ is orthogonal matrix : ...
1
vote
1answer
93 views

For which $\beta \in \mathbb{C}$ is the matrix $A=\bigl(\begin{smallmatrix} 0&1\\1&\beta \end{smallmatrix}\bigr)$ diagonalisable?

I have got a question refering to the following problem. Let $K=\mathbb{C}$. For which $\beta \in \mathbb{C}$ is this matrix diagonalisable? $$A=\pmatrix{0&1\\1&\beta}$$ I think that it is ...
2
votes
2answers
127 views

Area of an ellipse.

I need to find the area of the image of a circle centred at the origin with radius 3 under the transformation: $ \begin{pmatrix} 3 & 0\\ 0 & \frac{1}{3} \end{pmatrix} $ The image is the ...
1
vote
1answer
59 views

what represents this strange matrix [closed]

But I just encountered a very strange type of matrix. What is the name of this type of matrix? How can it be solved? The final result should be a $4\times 4$ matrix, but I only see $5$ elements on ...
1
vote
1answer
50 views

proof that a orthogonal endomorphism $f$ of $V$ exists, so that $f(W_1)=W2$. [closed]

$V$ is a finite dimensional euclidean vector space. $W_1$,$W_2$ are subspaces of $V$. $\dim(W_1)=\dim(W_2)$. How do you prove that an orthogonal endomorphism $f$ of $V$ exists such that $f(W_1)=W_2$.
5
votes
1answer
1k views

Monotone matrix

A real matrix $A$ is called monotone if $Ax\geq 0$ implies $x \geq 0$. If inverse of $A$ exists and is real, then prove that $A$ is monotone if and only if inverse of $A \geq 0$. ($x\geq 0$ means $x$ ...
4
votes
2answers
195 views

Determinant of matrices without expanding [duplicate]

Show that $$\begin{array}{|ccc|} -2a & a + b & c + a \\ a + b & -2b & b + c \\ c + a & c + b & -2c \end{array} = 4(a+b)(b+c)(c+a)\text{.}$$ I added the all rows but couldn't ...
1
vote
1answer
77 views

Finding the characteristic polynomial of $A^2$ given the characteristic polynomial of $A$

To find the characteristic polynomial of the matrix $A^2$, would I just compute $$(\lambda^2+4\lambda-5)^2 ?$$
2
votes
0answers
30 views

Inequalities on Matrix Minimax?

Suppose I have a matrix $M$. How can I get a good bound on the minimax quantity $$ \min_{i}\max_{j}M_{ij} $$ or variations thereof? Links to literature would be greatly appreciated. EDIT: I ...
3
votes
1answer
119 views

Suppose $AB=BA$ and $A^{1965}=B^{2015}=I$. Prove that $A+B+I $ is invertible.

Supppse $A $ abd $B $ are matrices, $AB=BA $ and $A^{1965}=B^{2015}=I $. Prove that $A+B+I $ is invertible. I want to prove that $(A+B+I)C=I $ I have no idea how to start. Can any one give some hint? ...
1
vote
1answer
2k views

How do you find angular velocity given a pair of 3x3 rotation matrices?

Let's say I have two 3x3 rotation matrices R1 and R2, each signifying rotation from the global frame to the local frame. I am also given the time difference t between these two matrices. How would I ...
1
vote
1answer
39 views

Cases of $A^2 = -I$. Why is there a contradiction when reusing this proof?

I had to prove that $\nexists ~A \in M_{3,3}(\mathbb R) : A^2 = - \mathbb I.$ I argued $$\iff A=-A^{-1}$$ $$\iff \det( A)=\det(-A^{-1})$$ $$\iff \det( A)=(-1)^n\det A^{-1}$$ $$\iff \det (A) + \det ...
6
votes
2answers
5k views

Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way? A quick search reveals that ...
2
votes
1answer
34 views

Let $T:U\rightarrow V$ be a linear map and suppose that $rank(T)=dim(U)=dim(V)=n$. Show that the are bases where the matrix is $I_n$

I found this problem that I cannot solve, but I believe is quite interesting. We have to state whether the statement is true or false. Let $T:U\rightarrow V$ be a linear map and suppose that ...
0
votes
1answer
164 views

Inverse Square Root Of Matrix

So let's say a matrix is A. Then how do you find A^-1/2? It seems to be different from finding the inverse of A. Could someone provide a simple example as ...
0
votes
1answer
31 views

$A$ is positive definite if and only if $Q$ is invertible for every choice of $Q$

Note that if $A \in M_{n \times n}$, $A^{\prime}$ denotes the transpose of $A$. I proved the following theorem already: $A$ is nonnegative definite if and only if there exists a square matrix ...
2
votes
1answer
39 views

Building matrices from eigenvalues

I saw a question some time ago, asking about the eigenvalues of the matrix $$A=\begin{pmatrix}5&-3&0\\-3&5&0\\0&0&2\end{pmatrix}$$ which were then shown to be ...
2
votes
1answer
24 views

Finding a base from matrice subspace

$U,W$ are sub-spaces of $M^\mathbb{R}_{2x2}$ $$U=Sp\left\{\begin{pmatrix} 1 & 2\\ 4 & 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & -1\\ 3 & 2 \\ \end{pmatrix}, \begin{pmatrix} 1 & ...
0
votes
0answers
26 views

Non negative irreducible matrix implies there is a strictly positive power.

How can I proof that a non negative irreducible matrix necessarily has a strictly positive power? By irreducible matrix i understand this http://mathworld.wolfram.com/ReducibleMatrix.html It looks ...
0
votes
0answers
41 views

How to simplify the kronecker product of four product

Suppose that $A$ and $B$ are $N\times N$ matrices, and $I$ is a $m\times m$ identity matrix, then here comes the kronecker product $$ K_1 = (I\otimes A)\otimes(I\otimes B). $$ I now wonder how can we ...
1
vote
2answers
121 views

Find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not.

I am trying to find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not. Have you got any ideas of easy examples? Thank you!
1
vote
1answer
87 views

How to show the following matrix is positive definite?

How to show the following matrix is positive definite. \begin{equation} \sum_{i=1}^n \Big[(d_i^Tp)^2\left\{\left( \begin{array}{c} d_i\\ A_ip \end{array} \right) \left( \begin{array}{c} ...
1
vote
1answer
50 views

matrix transformation - eigenvector

I am trying to understand eigenvectors. An Eigenvector is nothing more than a vector that points to some place. This pointing vector will then be invariant under linear transformations. Now my ...
0
votes
1answer
39 views

can use diagonal matrix in a formula to figure out how many characters would occur in all substrings of a string 's'?

Math experts - I'm working through a simple "big O" analysis of algorithms problem comparing two approaches to the longest substring problem. One approach is brute force: checking all possible ...
0
votes
0answers
29 views

Is it true that $\tilde{P} = D^{\dagger} P D$ has non-negative entries?

Consider a $n \times n$ stochastic matrix $P$ (i.e. non-negative rows sum to one). We are interested in the matrix $\tilde{P} = D^{\dagger} P D$, where $D$ is a $n \times k$ matrix which is ...
1
vote
1answer
32 views

Subspace of $\mathbb{R}^3$: Stuck on closed under addition

$$S=\left \{ \begin{bmatrix} x_{1}\\ x_{2} \\ x_{3} \end{bmatrix} ; x_{1}^{2}+x_{2}^{2}=x_{3}^{2} \right \}$$ Closed under addition: Let $\vec{y}=\begin{bmatrix} y_{1}\\y_{2} \\ y{3} ...
0
votes
1answer
34 views

finding angle and scalar c

matrix $$A = \pmatrix{ 4&-5\\5&4}$$ is standard matrix of a linear transformation from $R^2 \to R^2$ that consists of a rotation through an angle composed with multiplication by a scalar $c.$ ...
1
vote
3answers
37 views

Finding a nullspace of a matrix - what should I do after finding equations?

I am given the following matrix $A$ and I need to find a nullspace of this matrix. $$A = \begin{pmatrix} 2&4&12&-6&7 \\ 0&0&2&-3&-4 \\ ...
1
vote
3answers
54 views

Is any linear transformation with $\text{ker }(T)=\left\{\vec{0}\right\}$ an isomorphism?

I'm thinking no; for instance, $\exists \left\{\left.T:V\rightarrow W\right| \text{Im }(T)\neq W\right\}$. This seems counterintuitive, though. If such a $T$ with maximal rank exists, What would ...
0
votes
3answers
104 views

Prove that A is invertible if $A^2 - 4A -7I = 0$. [duplicate]

The $2 \times 2$ matrix $A$ satisfies $$A^2 - 4A -7I = 0,$$ where $I$ is the identity matrix. Prove that $A$ is invertible. I'm not sure how to do this. Help would be appreciated.
0
votes
1answer
70 views

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there… [duplicate]

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that Aw = v. Show that ${A}$ is invertible. I'm not sure how to do this.
1
vote
1answer
24 views

Finding a matrix by using hermitian

$A=\left[ \begin{array}{ccc} 4 & 0 & 0 \\ 0 & 1 & i \\ 0 & -i ...