# 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.

35,069 questions
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### How do I label the solution of a word problem using matrix multiplication?

The table on the left gives the birth and death rates (per million) by region. The table on the right gives the populations (in millions) in each region for a number of years. Use matrix ...
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### Has there been a rigorous analysis of Strassen's algorithm?

According to Wikipedia, Strassen's Algorithm runs in $O(N^{2.807})$ time. Has anyone seen a more rigorous analysis displaying constants, possibly in a specific language such as C or Java? I ...
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### Conjugacy class of a permutation from its matrix representation

Apologies if this is too basic, but given a permutation matrix $M$, is there any parameter or formula based on $M$ that gives the disjoint cycle decomposition, or at least the conjugacy class, of the ...
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### How many $n\times m$ binary matrices are there, up to row and column permutations?

I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations. If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
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### Proof - Square Matrix has maximal rank if and only if it is invertible

Could someone help me with the proof that a square matrix has maximal rank if and only if it is invertible? Thanks to everybody
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### Find matrices $A$ and $B$ given $AB$ and $BA$

Given that: $$AB= \left[ {\matrix{ 3 & 1 \cr 2 & 1 \cr } } \right]$$ and $$BA= \left[ {\matrix{ 5 & 3 \cr -2 & -1 \cr } } \right]$$ find $A$ and $B$.
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### Calculating axis angle from matrix without wrapping at PI

I need to get the axis angle from a matrix of the form: $$\begin{matrix} \cos \theta & -\sin \theta & tx\\ \sin \theta & \cos \theta & ty\\ 0 & 0 & 1\\ \end{matrix}$$ ...
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### shortcut for finding a inverse of matrix

I need tricks or shortcuts to find the inverse of $2 \times 2$ and $3 \times 3$ matrices. I have to take a time-based exam, in which I have to find the inverse of square matrices.
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### Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks

Let $A$ be a block upper triangular matrix: $A = \left( \begin{matrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{matrix} \right)$ where $A_{1,1} ∈ C^{p×p}$, $A_{2,2} ∈ C^{(n-p)×(n-p)}$ Show ...
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### Knowledge about Graph Spectral Theory and correlation between a graph Weighted Adjacency matrix and its eigenvalues

I know that this question is some sort of bridge between Informatics and Mathematics, not knowing the best place where to post this question, I opted for this place because of the type of answer I ...
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### Physical meaning of the null space of a matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it. E.g.: I think ...
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### Orthogonal Decomposition of A Matrix

I'm trying to follow/understand a research paper that I have, and well, it's been a while since I've done this kind of math. At this point I have an nxn matrix H and from that construct an (n-1)xn ...
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### Importance of matrix rank

What is the importance of the rank of a matrix? I know that the rank of a matrix is the number of linearly independent rows or columns (whichever is smaller). Why is it a problem if a matrix is rank ...
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### rotation matrix for particle track

Struggling with some equations here. I've got a particle I need to track in an absolute reference frame, but each step I move/rotate it relative to its own reference frame. I need to track it's ...
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### What's the fastest way to take powers of a square matrix?

So I know that you can use the Strassen Algorithm to multiply two matrices in seven operations, but what about multiplying two matrices that are exactly the same. Is there a faster way to go about ...
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### Positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than $1$ and off-diagonal entries positive but strictly less than $1$ necessarily positive-semidefinite?
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### A direct proof of the properties of the matrix of minors

Let $A$ be an invertible $n\times n$ matrix. Define the matrix of minors $\Delta(A)$ of $A$ to be the matrix whose $(i,j)$ entry is the determinant of the minor of $A$ with the $i$th row and $j$ ...
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### How to use parameterized solution to find solutions?

An electronics company produces three models of stereo speakers, models A, B and C, and can deliver them by station wagon, truck, or van. A station wagon holds $3$ boxes of model A, $3$ boxes of model ...
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### Trouble deriving the Harris Corner Detection

I just started studying a small paper about the Harris Corner Detection. The problem is I don't understand how step 7 is derived from step 6. In step 7 the expression is expanded in a way that we get ...
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### How to solve this problem using matrix

A company produces three combinations of mixed vegetables that sell in 1Kg package. Italian style combines 0.3kg of zucchini, 0.3kg of broccoli and 0.4kg of carrots. French style combines 0.6kg of ...
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### Invertible $N \times N$ matrix over ${\rm GF}(2)$ having on each row and column $N/2$ ones

As per the title, I'm looking for the name and for a way to construct a ${\rm GF}(2)$ square matrix of size $N$ with the following properties: All rows/columns should be linearly independent On each ...
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### Integer matrices with integer inverses

If all entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational. Now suppose that all entries of an invertible matrix $A$ are integers. Then it's not ...
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### Inverting $(A + D_p)$ efficiently for many diagonal $D_p$ (and $A$ SPD)

At one step of an algorithm (i=1: something big) I have to draw a vector from the normal distribution $X_p^{(i)}\sim N((A^{(i)} + D^{(i)}_p)^{-1}m, (A^{(i)} + D^{(i)}_p)^{-1})$ for $p=1:P$, with a ...
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### From a triangular matrix to a symmetric square matrix

I am a computational physics postgrad student, working with libraries like ATLAS and MAGMA. I have a matrix which is upper-triangular, and is the result of a Cholesky decomposition. I need to ...
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### Matrix and Binomial Coefficients

Considering the construction of a matrix as follows. The $n$th row in the matrix is filled with the coeffcients of $x^r$ in the expansion of $(1+x)^n$ from the columns $2n$ to $3n$ inclusive and ...
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### Normalizing the Eigenvalues so their sum is equal to min(n, p)?

If I compute the eigenvalues and eigenvectors using numpy.linalg.eig (from Python), the eigenvalues returned seem to be all over the place. Using, for example, the ...
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### connection between graphs and the eigenvectors of their matrix representation

I am trying to learn graph theory and the linear algebra used to analyse graphs. The texts I have read through have lots of lemmas and theorems proved. The proofs are convincing but I fail to see the ...
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### Question about the standard basis's relationship to the identity matrix

I'm in the process of reading my first Linear Algebra textbook, and was just wondering...Is the standard basis of a vector space in n dimensions equivalent to the row space of the n x n identity ...
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### Showing that the path-connected component of the identity matrix in a subgroup of $GL_n(\Bbb R)$ is a normal subgroup

Let $M(n;\mathbb{R})$ denote the set of all $n \times n$ matrices with real entries (identified with $\mathbb{R}^{n^{2}}$ and endowed with its usual topology) and let $GL(n;\mathbb{R})$ denote the ...
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### Verifying the distribution of an ensemble of random matrices

Suppose one is trying to devise a method to generate random matrices with a certain distribution. How does one verify that the generated matrices follow the desired distribution? In particular, I am ...
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### Help me spot the error?

I have a determinant to expand which is $$\triangle = \begin{bmatrix} p& 1 & \frac{-q}{2}{}\\ 1& 2 &-q \\ 2& 2 & 3 \end{bmatrix} = 0$$ But when I am expanding the ...
How do i expand and simplify the vector expression $(\vec{a}-\vec{b})^T.(\vec{a}-\vec{b})$ = ? And if there are matrices A and B instead of vectors a and b, how do i multiply and simplify this ...
### Compactness of the set of $n \times n$ orthogonal matrices
Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.