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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How to tell if two matrices are similar?

Two n-by-n matrices A and B are called similar if $$ \! B = P^{-1} A P $$ for some invertible n-by-n matrix P. Similar matrices share many properties: Rank Determinant Trace Eigenvalues (though the eigenvectors will in general be…
Tim
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Convergence of a sequence of eigenvectors (nonnegative matrix)

Let $A$ be a $n\times n$ matrix with coefficients in $ [0,1] $. Let $ B $ be the matrix filled up only with the value $ \frac{1}{2} $: $$B = \begin{pmatrix} \frac{1}{2} & \dots & \frac{1}{2} \\ \vdots & \ddots & \vdots \\ \frac{1}{2} & \dots &…
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Write $(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$ as a sum of (three) squares of quadratic forms

The quartic form $$(x^2 + y^2 + z^2)^2 - 3 ( x^3 y + y^3 z + z^3 x)$$ is non-negative for all real $x$, $y$, $z$, as one can check (with some effort). A theorem of Hilbert implies that there exist quadratic forms $Q_1$, $Q_2$, $Q_3$ so that $$(x^2…
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When is a matrix triangularisable?

When is a matrix triangularisable? I don't seem to find much online regarding the triangularisability of matrices. What should I look for to prove if a matrix is triangularisable? What are the implications on its eigenvalues eigenvectors and…
Leroy
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Computing determinant without expansion

$$\begin{align}\mathrm D &= \left|\begin{matrix} (b+c)^2 & a^2 & a^2 \\ b^2 & (a+c)^2 & b^2 \\ c^2 & c^2 & (a+b)^2 \end{matrix}\right|\\ &= (a+b+c)\left|\begin{matrix} b+c - a & a^2 & a^2 \\ b - a -c & (a+c)^2 & b^2 \\ 0 & c^2 & (a+b)^2…
user8277998
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Show that $(AB-BA)^2=O_2$

Let $A,B$ be two $2 \times 2 $ matrices with real elements and $a,b \in \Bbb R$ such that $a^2 \ne b^2$ and $A(A-aB)+B(B-bA)=O_2$. Show that $(AB-BA)^2=O_2.$ MY TRY: Using Hamilton-Cayley Theorem, the problem reduces to showing that $\det (AB-BA)=0$…
M. Stefan
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Relations between matrix norm and determinant

I was wondering whether there is a way to obtain the determinant of a matrix out of its norm (when the matrix is regular otherwise it is not true). If $A$ is a square matrix of dimension $n\geq 1$, and $\det A\neq 0$, do we have something…
Martingalo
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Can you find a minimal polynomial of $A^n$ if you know the minimal polynomial of $A$?

Can you find a minimal polynomial of $A^n$ if you know the minimal polynomial of $A$? I'm talking about minimal polynomials of matrices. I'm asking in the general sort of way, I know that in some cases you can use algebraic tricks or some other…
ante.ceperic
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Help understand an inequality in a proof

Assume standard inner product and 2-norm. $A$ is any symmetric real-valued $n\times n$ square matrix. $V_k$ is an $n\times k$ matrix whose columns are orthonormal. $T_k$ is a matrix such that it has the following relation with $A$ and $V_k$:…
River
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Matrix norm of Kronecker product

Is it true that $ \| A \otimes B \| = \|A\|\|B\| $ for any matrix norm $ \|\cdot \| $? If not, does this identity hold for matrix norms induced by $ \ell_p $ vector norms?
Rob
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What are the diagonals in the matrix called?

How correctly to designate these diagonals? They are highlighted in different…
Maria
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If $p+q+r=0$, find the value of the determinant

If $p+q+r=0$, prove that the value of the determinant $$ \Delta= \begin{vmatrix} pa & qb &rc \\ qc & ra &pb\\ rb& pc & qa \\ \end{vmatrix} =-pqr \begin{vmatrix} a & b &c \\ b & c &a\\ c& a & b \\ \end{vmatrix}$$ My Try:Since $p+q+r=0$ we…
Ekaveera Gouribhatla
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Does $\mathrm{adj}(A)=\mathrm{adj}(B)$ imply $A=B$?

If $A$ and $B$ are any two square matrices of same order and if $\mathrm{adj}(A)=\mathrm{adj}(B)$, does it imply $A=B$? I am pretty sure if $A$ and $B$ are invertible and if $A^{-1}=B^{-1}$, then $A=B$. So is it true for Adjoint?
Ekaveera Gouribhatla
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Prove that matrix can be square of matrix with real entries

Prove that matrix \begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix} can be square of matrix with all real entries. I have found one such matrix to be \begin{bmatrix}1&0&0\\0&1&-1\\0&2&-1\end{bmatrix} but is there an elegant way to do it without…
Mathematics
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Definiteness of a general partitioned matrix $\mathbf M=\left[\begin{matrix}\bf A & \bf B\\\bf B^\top & \bf D \\\end{matrix}\right]$

If $\mathbf M=\left[\begin{matrix}\bf A & \bf b\\\bf b^\top & \bf d \\\end{matrix}\right]$ such that $\bf A$ is positive definite, under what conditions is $\bf M$ positive definite, positive semidefinite and indefinite? It is readily seen that…
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