Tagged Questions

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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Proof related to matrix with if and only if condition

Suppose $A$ and $B$ are matrices such that $AB$ and $BA$ are defined. Prove that $$(A+B)^2=A^2+B^2+2AB\quad\text{if and only if}\quad AB=BA.$$ Someone help me with this.
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How to approach specific dimensionality reductions?

I'm having difficulty in rationalizing dimensionality reduction (I've used other sources), and I would appreciate it if someone could help me out with a specific example. Given an $M \times M$ PCA ...
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Making a matrix diagonal with its eigenvectors

I'm trying to make my matrix diagonal. this is my matrix (for matlab and octave) ...
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Prove these matrix-vector products are linearly dependent/independent.

I have two statements that I wanto to prove or disprove. Let $A \in \Bbb R^{n \times n}$ and $b_1,\ldots,b_n \in \Bbb R^n$ be linearly independant. Then, $Ab_1,\ldots,Ab_n$ are also linearly ...
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Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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How to prove Cholesky decomposition for positive-semidefinite matrices?

According to Cholesky decomposition $A$ is a Hermitian positive-definite matrix if and only if $A=T^*T$ for some upper triangular matrix $T$. When $A$ is positive-semidefinite we have such ...
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matrix inequality proof [completion of squares]

Can someone help me to prove this? $\begin{bmatrix} 0 & B^\top W^\top \\ WB & 0 \end{bmatrix} \leq \begin{bmatrix} B^\top Q B & 0 \\0 & W^\top Q^{-1}W \end{bmatrix}$ with $Q$ ...
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Proving generalized form of Laplace expansion along a row - determinant

Definition: Let $A$ be an ($n \times n$)-matrix. Let $M_{ij}$ denote the matrix acquired from $A$ by deleting row $i$ and column $j$. For $n \geq 2$ we define the determinant of $A$ inductively as \...
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Prove that $\dim(V)$ is even

Let $V$ be a finite dimensional vector space. Let $A_1,A_2: V\rightarrow V$ be commuting linear operators such that $A_1+A_2=-I$ where $I$ is the identity operator. Also $A_1,A_2$ have no negative ...