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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2
votes
1answer
36 views

Determinant of matrices without expanding

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 ...
0
votes
1answer
39 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 $$(\lambda^2+4\lambda-5)^2 ?$$
1
vote
3answers
56 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
1answer
17 views

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

I need to find the projection of $\v b = (1,1,1,1)$ 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.
1
vote
0answers
21 views

Why does $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold?

The following problem is from Taylor's PDE I. I do not get why $D \det(I)B=\frac{d}{d t}\det(I+t B)|_{t=0}$ hold. Since $\det(I)=1$ and $D$ is $n\times n$ matrix, the left side seems to be a matrix, ...
4
votes
1answer
72 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? ...
0
votes
0answers
19 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 ...
0
votes
1answer
27 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 ...
-2
votes
0answers
7 views

Calculate the matrix of a lineal aplication with some information

‪If f:(Z7)3 à(Z7)3 is the only lineal application with Ker(f)= and V2={(1,0,1), (1,1,0)} = L[(1,0,1),(1,1,0)] where V2 is the subspace associated to the proper value 2. Calculate the matrix ...
2
votes
1answer
17 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 ...
4
votes
1answer
35 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
1answer
22 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 ...
-3
votes
0answers
14 views

Calculate coordinate transformation matrix [on hold]

I have problem at the following poblem: Calculate coordinate transformation matrix A in the base [1; x; x2]. In the space $R_2 [x]$ of real polynomials highest rate? D is given a scalar product $A: ...
1
vote
1answer
29 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 ...
0
votes
0answers
30 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 $|||⋅|||$ ...
0
votes
0answers
9 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 ...
1
vote
1answer
15 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
24 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
54 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
23 views

Matrix irreducibility. Strongly connected graph

I have this theorem from Combinatorial Matrix Theory written by Richard A. Brualdi and others. Let $A$ be a matrix of order $n$. Then $A$ is irreducible if and only if its digraph $D$ is strongly ...
1
vote
1answer
18 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
0answers
20 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 ...
0
votes
1answer
17 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
11 views

Why is a BCCB matrix completely specified by its first column?

It is claim in some literature that a Block Circulant with Circulant Block (BCCB) matrix is completely specified by its first column.(e.g. here ) But I have a contradictory example: Let $c = [1, 2, ...
1
vote
1answer
31 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
2answers
21 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 $c.$ ...
2
votes
2answers
198 views

Show that 1 and -1 are the only eigenvectors of this linear transformation

Define $T: M_{n\times n}\to M_{n\times n}$ by $T(A):= A^t$. Note that $T$ is a linear transformation. Show that $1$ and $-1$ are the only eigenvalues of $T$. Let $\lambda$ denote an eigenvalue ...
2
votes
3answers
114 views

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$?

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$? I read a book a very long time ago where it said something like this: Given a system of two equations with two ...
0
votes
1answer
24 views

Using Gauss elimination to check for linear dependence

I have been trying to establish if certain vectors are linearly dependent and have become confused (in many ways). when inputting the vectors into my augmented matrix should they be done as columns or ...
0
votes
3answers
29 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 \\ ...
0
votes
3answers
62 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
20 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.
2
votes
3answers
57 views

Find an arbitrary power of a lower triangular matrix of size $3\times 3$

Let $F$ be a field and let $A=\begin{bmatrix}a&0&0\\1&a&0\\0&1&a\end{bmatrix}\in\mathscr{M}_{3\times 3}(F)$. Show that ...
-2
votes
0answers
11 views

How to find proper parameter t to diagonalise matrix [on hold]

Find the proper parameter $t$ to diagonalise this $3 \times 3$ matrix. $\left|\begin{array}{ccc} 1 & t & 25 \\ 0 & t & t+1 \\ 0 & 0 & -1 \end{array}\right|$ How do I solve ...
1
vote
1answer
95 views

Proving that matrix in equation is invertible

The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible. I have tried to solve it like a quadratic, but ...
-2
votes
1answer
41 views

What is the 5th root of the stated matrix? [on hold]

A $3\times3$ matrix $\left[\begin{array}{ccc}1&2&0\\-1&-2&0\\3&5&1\end{array}\right]$ What should I do ? Many thanks
0
votes
2answers
24 views

Orthogonality and projections

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 ...
3
votes
1answer
25 views

How to prove this identity involving characteristic polynomials on both sides?

Suppose $A\in \Bbb C^{m\times n},B\in \Bbb C^{n\times m},m\ge n$, prove: $$\det(\lambda I_m-AB)=\lambda^{m-n}\det(\lambda I_n-BA)$$ I don't want to get into nasty determinant calculation. Instead, I ...
1
vote
1answer
17 views

Finding a matrix by using hermitian

$A=\left[ \begin{array}{ccc} 4 & 0 & 0 \\ 0 & 1 & i \\ 0 & -i ...
0
votes
2answers
32 views

what do eigenvalue & eigenvector of $4\times4$ matrix represent?

What do we get when calculating the eigenvalue and eigenvector of a $4\times4$ matrix? What do those values actually represent?
1
vote
2answers
21 views

Let $A$ be a complex $2$ by $2$ matrix having distinct eigenvalues $a, b$. Show that $A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A - aI)$.

Let $A\in\mathscr{M}_{2\times 2}(\mathbb{C})$ be a matrix having distinct eigenvalues $a\neq b$. Show that, for all $n > 0$, \begin{equation*} A^n =\frac{ a^n}{a - b}(A - bI) + \frac{b^n}{b - a}(A ...
2
votes
5answers
38 views

Proving any vector in $\Bbb R^n$ can be written on the form $x = u + v$

I'm having a hard time understanding the solution of this exercise. The exercise says: Let A be an $n\times n$ matrix so that $$A^2 = A$$ Show that every vector $x$ in $\Bbb R^n$ can be written as ...
1
vote
2answers
30 views

How can you find a matrix given you know its kernel/nullspace?

Suppose we are given that $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$, and also that $\ker\phi$ is the span of $\{\begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 ...
0
votes
1answer
61 views

vectors and matrix problem [on hold]

Let $A$ be a $5$x$5$ matrix, with the third column of $A$ represented by $a_3$, and let $b$ be a $5$x$1$ non-zero column vector. Suppose that the matrix equation $Ax=b$ has a unique solution x, with ...
0
votes
0answers
38 views

Matrix problem, subspace

Suppose you are given a matrix A and have calculated an echelon form R of A. (Note: R is not assumed to be in reduced row echelon form.) Which of the following statements must be true? (Select all ...
-2
votes
1answer
24 views

A basic question about eigenvalue

Suppose a symmetric matrix $A$ is of dimension $N \times N$. Then the largest eigenvalue of $A$ is equal to $\max_{i} \sum^{N}_{j=1} |A_{ij}|$. Is this statement true? If so, how shall I show it ...
0
votes
2answers
30 views

Is the following set of vectors in $\Bbb R^3$ linearly dependent?

I am using Anton's Elementary Linear Algebra book (8e) and trying to do exercise set 5.3, question 2a It gives the vectors $(4,-1,2)$, $(-4,10,2)$ and asks if they are linearly dependent . My final ...
3
votes
1answer
21 views

Determinant of block matrix with commuting blocks

I know that given a $2N\times 2N$ block matrix with $N\times N$ blocks like $\mathbf{S} = \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ we can calculate ...
0
votes
1answer
49 views

Showing a $2\times2$ matrix is invertible

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that ${A} {w} = {v}.$ Show that ${A}$ is invertible. I have no idea on how ...
2
votes
1answer
31 views

Checking psd-ness of matrix

I have the following problem and don't know how to proceed... I want to check if \begin{equation} \frac{1}{2}(B^\top A^\top A + A^\top A B) - \frac{1}{4}B^\top A^\top A A^\top (AA^\top ...