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Mathematics Stack Exchange

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.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).

For questions specifically concerning matrix equations, use the tag.

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Permutation matrices

Let $\mathscr{M}$ be the set of all $n\times n$ matrices having entries $0$ and $1$ in such a way that there is one $1$ in each row and column. (a) If $M\in\mathscr{M}$, describe $AM$ in terms of the rows and columns of A. (b) If…
creative
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Why can I not write every $N \times M$ matrix as multiplication of an $N \times 1$ and a $1 \times M$ matrix?

My intuition says I simply can't express $N \times M$ independent variables in terms of $N+M$ variables but how can I show that?
Atilla Filiz
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How to compute the group inverse of $M+aI_n-\frac{a}{b}J_{n\times n}$?

For a square matrix $M$, the group inverse of $M$, denoted by $M^\#$, is the unique matrix $X$ such that $MXM=M$, $XMX=X$ and $MX=XM$. Given an $n \times n$ matrix $M$, let $I_n$ denote the identity matrix of size $n$ and $J_{n\times n}$ denote the…
Ralph
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Making it positive semdefinite

What conditions( on $a$ and $b$) I need to impose on the following matrix to make it positive semidefinite? $$A=\begin{pmatrix}a&b\\b&0\end{pmatrix}.$$ Thanks in advance.
Neon
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Matrix multiplication verification

So I have this ought to be simple example in my lecture notes but I just can't wrap my head around this guys solution. I understand how there are $n$ multiplications but how are there $n$ additions for each element? Given a $3\times3$ multiplied by…
Cameron
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Prove or disprove that AB=AC $\implies$ B=C

I proved it as follows but I'm not so sure about it. A, B and C are square matrices of the same order. Assume $ B \neq C $ $$ AB \neq AC$$ $$ B \neq C \implies AB \neq AC$$ $$ \neg ( AB \neq AC) \implies \neg ( B \neq C ) $$ $$AB =AC \implies B=C…
S.Dan
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Gerschgorin's Disks of a 3x3 complex matrix

How to estimate the radii of Gerschgorin's Disks from a 3x3 square complex matrix? Is the disks just simply center at the diagonal elements with radius equal to the sum of the absolute values of other elements of the same row?
Putterboy
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Stuck at Smith normal form

Can somebody help me with the Smith normal form of this matrix? I know what it should be, but I get stuck at some point. Can you show how to take it from the point I'm stuck? This is the matrix: original matrix…
Andrei Stirb
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Sign of a Quadratic Form $X'AX + B'X+C$

We know that a quadratic form: $X'AX$, in the variable $X$, the matrices having their appropriate dimensions, is negative for all $X$, iff the matrix $A$ is negative definite. Can this result be extended for the case when the quadratic expression…
Anghri
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Matrix theory: Solution of a system

$ A = \left[ {\begin{array}{cc} 1 & 2 & 3 \\ 3 & 7 & 7 \\ 1 & 1 & 3 \end{array} } \right] $ $ B = \left[ {\begin{array}{cc} 1 & 0 & 3 \\ 2 & 1 & 7 \\ 3 & 2 & 9 \end{array} } \right] $ if $ b = \left[ {\begin{array}{cc} b_1 \\ b_2 \\ b_3 \end{array}…
user3542501
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Can zero rows in matrices be ignored in calculations of matrix products?

I understand that when calculating the product of 2 matrices you need to account for the dimensions. But when there is an empty row in one of the matrices, why does it need to be accounted for? What are the implications if you ignore that row before…
Jade
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Matrices: $AB=0 \implies A=0 \text{ or } \ B=0$

When A and B are square matrices of the same order, and O is the zero square matrix of the same order, prove or disprove:- $$AB=0 \implies A=0 \text{ or } \ B=0$$ I proved it as follows:- Assume $A \neq O$ and $ B \neq O$: then, $$ |A||B| \neq 0…
S.Dan
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Quick way to determine congruences

Is there a quick way to determine if a $2\times 2$ matrix, $M\in M_2(\mathbb R)$, is congruent to $I_2$ over $\mathbb R, \mathbb C, \mathbb Q$? Without explicitly finding the matrices $P\in M_2$ s.t. $I_2=P^TMP$? Brainstorm: Perhaps usig ranks and…
alec
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Possible known ways to improve Back Substitution

I am quite new to this field and just implement algorithms. I am currently using back-substitution as a way to invert a lower triangular matrix. I would like to ask if there are known ways to improve that back-substitution algorithm,in terms of…
Rizias
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Matrix Decomposition

Possible Duplicate: square root of symmetric matrix and transposition If I have a matrix K that is symmetric and positive definite, is there a way to decompose it into matrices K = A*A'? I had a look at…
Projectile Fish
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