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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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1answer
38 views

Range space of matrices over $\mathbb{Z}$

Let A and B be $m \times n$ matrices over $\mathbb{Z}$ such that $B=PAQ$ for some invertible matrices P and Q. Then can we tell that Range space of A is same as that of the range space of B when A ...
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2answers
95 views

Determinant of a $3 \times 3$ matrix [closed]

I have a matrix of order $3 \times 3$. When I take its determinant it give 1700 from all the rows and columns except row 2. I don`t know whats going on $$\begin{bmatrix}1&20&0\\0&0&10\...
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votes
2answers
111 views

Finding basis and nullspace

The question is as follows : If P is the plane of vectors in $$R^4$$ satisfying $$x_1+x_2+x_3+x_4=0$$ write a basis for P perp. Construct a matrix that has P as its nullspace So below is my approach :...
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votes
1answer
573 views

rank of a vector space [closed]

I read this sentence in a report concerning in symmetric cone programs: "Let J be a Euclidean Jordan algebra with dimension n, and rank r." I know what the rank of a (matrix) is.. does the rank here ...
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votes
1answer
437 views

MAthematical notation for sorting submatrix and replacing it back

I need help in expressing the following paragraph in mathematical form as much as possible. I have a matrix $A$ which is $N\times M$. For each element of $A$, $A(i,j)$, I consider a submatrix of $A$ ...
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votes
2answers
58 views

Condition check for matrices

If $A= \begin{bmatrix}2a & 2b \\ 2c & 0 \\ \end{bmatrix}$ and $B=2\begin{bmatrix}a & b \\ c & 0 \\ \end{bmatrix}$, then how is $A=2B$ ? Also, how is this possible? $\begin{vmatrix}B\...
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votes
2answers
2k views

How to find a unique solution, infinite solution and no solution for this matrix. [closed]

The question on my page is For what value(s) of k does the system have, no solutions, a unique solution, and infinitely many solutions? All help is appreciated! Thanks in advance
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1answer
34 views

Linear Algebra: If a is a $5\times 8$ matrix which of the following are true? [closed]

Can someone tell me which of the following are correct and explain why that is true? To me they are all false, but I am unsure.
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2answers
44 views

0 as approximate eigenvalue of a matrix [closed]

i got a problem that i cant solve. And i would be grateful for some help. Given the matrix $ X = \begin{pmatrix} 0 & 1 & & \\ -1 & 0 & \ddots & \\ & \ddots & \ddots ...
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1answer
39 views

Find coordinate vector in matrix vector space

How do I do this question? I don't understand the notation that describes B what is the superscript ij? what is E?
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2answers
34 views

Proof there isn't a vector u such Su=u where S is the rotation transformation in R2 [closed]

We have the rotation matrix \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{pmatrix} Proof: there isn't exist a vector $u \in\ {\mathbb{R}^2}$ ($u\neq0$) such $...
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votes
1answer
51 views

Conjugate matrices transposed? [closed]

If we denote $A^*=\left (\overline{A} \right)^T$ ( as in $\overline{A}$ transposed) as the conjugate matrix of $A$ why is the conjugate transposed or why does it have to be?
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1answer
79 views

How do I reasoning matrix of size n to get the big-o?

The following very simple algorithm calculates the determinant of a matrix in a recrusive fashion with no optimization at all. Count how many operations (+, -, * and /) are done for a matrix of size n ...
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1answer
35 views

matrix with two unknowns

I am to calculate the value of this matrix $$ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & a \\ 1 & b & 1 \end{bmatrix} $$ I do a basic ...
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1answer
278 views

Quadratic forms matrices

Let $$Q(x,y,z) = – 2x^2 + 6xy + 8y^2 + z^2.$$ Find the symmetric matrix associated with this quadratic form. Use the determinant method to determine whether the quadratic form is positive definite, ...
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1answer
53 views

Finding the unknown matrix X.

I'm practicing for my upcoming exam, and I can't find a way around this part of the question, Part a) "Given that A= (2 1 -2 5) , find the inverse of the matrix A+I, where I is the identity ...
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1answer
62 views

Column operations on matrix product

We have an invertible $n \times n -$matrix $A=[a_1,\ldots,a_n]$ and an invertible $n \times n -$matrix $B=[b_1,\ldots,b_n]$. We can with column operations turn both matrices into diagonal form: $A \...
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1answer
53 views

A 3D curve correlation

Forgive me if its too basic, but i am looking to read some materials about a subject in which i don't know its name/field. So what we need to do, is to get a 3 axises curve, with unknown shape, that ...
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2answers
486 views

Computationally more efficient technique than Singular Value Decomposition.

I am working on a mathematical project where I have to decompose a given matrix into two or more matrices. Presently I am using Singular Value Decomposition (SVD) for it. I came to know that SVD is a ...
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votes
1answer
223 views

How polynomials are represented in matrix form for Univariate Polynomial. [closed]

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
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votes
1answer
416 views

Solving matrix using Gaussian elimination and a parameter

$\begin{bmatrix} x_{1} & 2x_{2} & & & ax_{5} & x_{6} & = & -2 \\ -x_{1} & -2x_{2} & & & (-1-a)x_{5} ...
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1answer
270 views

Basis for Column Space

If the RREF of a matrix is the identity matrix, would the standard basis be a basis for its column space? And would the theorem that says a basis for the column space is the corresponding columns with ...
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1answer
179 views

Write matrix as a linear combination of polynomials

I can't figure out how to solve this:
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1answer
41 views

Convenient representation for matrix equation through rvec/devec operator and kronecker

I have matrix equation in the form $y = Ax$, where $y=\begin{pmatrix} y'_1 \\ y'_2 \\ \vdots \\ y'_M \\ y''_1 \\ y''_2 \\ \vdots \\ y''_M \end{pmatrix}; $$ and, $$ x = \begin{pmatrix} x'_1 \\ x'...
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1answer
305 views

Given a Adjacency Matrix Table, complete an adjacency List

I'm given the table: (An Adjacency matrix of a graph) $$\begin{array}{c|c|c|c|c|c|c|c|} & \text{a} & \text{b} & \text{c}& \text{d}& \text{e}& \text{f} & \text{g}\\ \...
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votes
1answer
119 views

$A$ is a $n\times n$ matrix and there exists a unique matrix $B$ such that $AB=I_n$. Prove that $BA=I_n$ and $B=A^{-1}$. [duplicate]

$A$ is a $n\times n$ matrix and there exists a unique matrix $B$ such that $AB=I_n$. Prove that $BA=I_n$ and $B=A^{-1}$. I have no idea how $BA=I_n$ and $AB=I_n$, one implies another. Please help me ...
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votes
1answer
48 views

What does it mean if, for a real input $x, x^T x$ is a singular matrix? [closed]

I was trying to find a least squares solution using the normal equation. I created an 'input' vector $x$ with a fixed coefficient times $i$ plus a random number. As far as I know, for a $x^T * x$ to ...
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votes
2answers
1k views

Finding Eigenvalues of a 3x3 Matrix INVOLVING LAMBDA

Need some help with determinants involving eigen's. I understand the steps used in the process below, but I don't understand how my teacher knew that he had to do those steps to get a nice 0 row with ...
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votes
1answer
94 views

Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix.

Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix. I'm not sure how to do this, any solutions/hints are greatly appreciated....
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1answer
50 views

Computing the variance of a matrix [closed]

If $X$ and $Y$ are independent random variables and we let $x=\begin{bmatrix}X+2\\3X\\3Y-1\end{bmatrix}$ How do you define a variance-like characteristic of a random vector variable? Unfortunately, ...
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1answer
57 views

Calculation roots of determinant of matrix polynomial

Let $A,B \in {\mathbb{C}^{n \times n}}$ and ${t_0} \in \left( {0,1} \right)$ if fix. Suppose $N = \left\{ {x \in \mathbb{C}:\det (({A_2} + {t_0}{B_2}){x^2} + ({A_1} + {t_0}{B_1})x + ({A_0} + {t_0}{...
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votes
1answer
39 views

Eigenvalues of Decomposition

Let $P= P^{\top} \in \mathbb{R}^{n \times n}$ be positive definite. Prove that there exists a diagonal, invertible matrix $D \in \mathbb{R}^{n \times n}$ such that the matrix $$ D^{\top} \, P \, D $$...
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1answer
81 views

Proving a matrix property of unnormalized laplacian matrix of a graph

Consider the symmetric real $n\times n$ matrix $W$. Define $d_i=\sum_{j=1}^{n}w_{ij}$. Define $D$ as the diagonal matrix with $d_i$ as its diagonal entries. Define $L=D-W$ as the laplacian matrix. Now ...
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votes
2answers
198 views

By using the properties of determinant show that

$$\begin{vmatrix}1&a&a^2\\ 1&b&b^2\\ 1&c&c^2\end{vmatrix}=(b-a)(c-a)\begin{vmatrix}1&a&0\\ 0&1&b\\ 0&1&c\end{vmatrix}$$ I have been trying to solve ...
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1answer
148 views

matrix multiplication questions [duplicate]

$A$ and $B$ are two matrices, when is $(A-B)(A+B)=A^2 - B^2$
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1answer
298 views

Inverse of the sum of the inverse of 2 non-invertible matrices

Given that the following square matrices are non-invertible: $\bf A$, $\bf B$, and (A+B) UPDATE: Assume $\bf (A+B)$ is invertible. and given that $\bf (A+I)$, $\bf (B+I)$, and $\bf [(A+I)+(B+I)]...
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votes
1answer
172 views

SVD proof to If $A$ is of full rank, then $A^{*}A$ is of full rank

Provided $A$ is a full rank matrix $\in\mathbb{C^{m\times n}}$, then $A^{*}A$ is of full rank. Suppose $m\gt n$. There is a solution to this problem: solution link, and the top solution makes sense ...
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votes
1answer
120 views

Column vector of simultaneous equaations' solution

Struggling with some basics of Linear Algebra. Please help. Let's restrict the discussion to 2D space & consider the following simultaneous equations: $2x + 3y = 8, x + 2y = 5$ I understand ...
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votes
2answers
33 views

Question on how a matrix is calculated from an example [closed]

I have the following laplacian matrix given to me in a textbook. In the textbook, the matrix calculation is always done from the 3 x 3 matrix (the methods I learnt makes me cut the matrix further ...
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votes
2answers
108 views

$LU$ Decomposition of antidiagonal matrix

I cannot find the $LU$ decomposition of anti-diagonal matrix $$\begin{bmatrix} 0 &0 &0 &1 \\ 0 &0 &2 &0 \\ 0 &3 &0 &0 \\ 4 &0 &0 &0 \end{bmatrix}.$$ ...
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votes
1answer
83 views

Number of unique variable solutions of an incomplete matrix

Consider the simple linear system: $$ 3 x_1 + 2x_2 = 4 \\ 5x_3 = 9 $$ and the corresponding matrix form Ax=b: $$ \left( \begin{array}{ccc} 3 & 2 & 0 \\ 0 & 0 & 5 \end{array} \right) ...
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votes
1answer
53 views

Dimension of Vector Spaces

Can anybody help me finding out the dimension of the vector spaces: A: A is $n\times n$ real upper triangular matrices. A: A is $n\times n$ real symmetric matrices. A: A is $m\times n$ real matrices.
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1answer
52 views

How to find the matrix of the transformation relative to the basis?

Let $T:P_2\to P_2$ be the linear operator defined by $$T(a+bx+cx^2)=(3a+2b+4c)+(2a+2c)x+(4a+2b+3c)x^2$$ Find the matrix of the transformation $T$ relative to the basis $B=\{1,x,x^2\}$.
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votes
1answer
71 views

Why do the columns of the inverse of a matrix (defined as a linear operator) form an orthogonal basis in an inner product space?

Let V be a vector space over C and W be an inner product space over C with inner product <., .> and T:V --> W be a linear transformation. Find an orthogonal basis for V = R^3 with the inner product ...
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votes
2answers
97 views

Converting a second order n x n system into a first order 2n x 2n system

Say I have the following second order 7 x 7 system of equations: $x_1'' = 10(x_2- x_1- 1)$ $x_2'' = 10(x_3- 2x_2+ x_1)$ $x_3'' = 10(x_4- 2x_3+ x_2)$ $x_4'' = 10(x_5- 2x_4+ x_3)$ $x_5'' = 10(x_6- ...
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votes
1answer
29 views

Null space and Matrix equations

http://studyguide.pk/Past%20Papers/CIE/International%20A%20And%20AS%20Level/9231%20-%20Further%20Mathematics/9231_s03_qp_1.pdf I would like to know the method to answer question 8. I have been having ...
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votes
1answer
119 views

How to prove the equality of two matrix expressions

I am new to linear algebra and my question maybe too simple. I have a n-by-m matrix $D$ that its columns have unit L2 norm. Let $D_a$ be a sub-matrix of $D$ composed by some columns of $D$. I need to ...
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votes
1answer
16 views

Let $S$ be nonsingular size $n$ matrix. Show that the matrix $S^-1AS$ has the same characteristic polynomial as $A$.

I was thinking to go this way: $|S^-1AS - LI|$ = $|S^-1AS - LS^-1S|$ = $|S^-1|$*$|AS - LS|$ = $|S^-1|$*$|A-LI|$*$|S|$ = $|A-LI|$ Is this right?
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1answer
87 views

Perform the following block multiplications

i understand this process. but i do not understand that how may i calculate this. kindly tell me.
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votes
1answer
963 views

Transformations between two coordinate systems on a rigid body

I have two coordinate frames, A and B, which are rigidly attached to each other on a body. This body then translates and rotates, such that A starts at A1, and moves to A2, and B starts at B1, and ...