# Tagged Questions

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### The Matrix Inversion Lemma: the General Case

I find it is hard to understand the application senario of the Matrix Inversion Lemma in non-special cases. Suppose I already computed $A^{-1}$ and want to find $\left(A+UCV \right)^{-1}$. The Matrix ...
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### Extending a set of vectors to a basis by picking from a given basis

I have a linear independent set ${\cal K}=\{v_1,\dots,v_{k-d}\}\subset\mathbb{R}^k$. I'd like to find $\cal W=\{w_1,\dots,w_d\}$ such that $\cal K\cup W$ is a basis for $\mathbb{R}^k$. To do this, ...
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### Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
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### Understanding of a formula with matrix summation

We are a quite a few students at in the class struggling to compute this would anyone be able to help? Also note --> ' means transpose. Sorry for the misunderstanding, when I said that it is not ...
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### Proving the equality: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$ where $P=(A-BD^{-1}C)^{-1}$

I am trying to prove the following equality that I need to use as an intermediate step to solve one of my problems. The equality is the following: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$, where ...
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### How to express a matrix as a product of two symmetric matrices?

Let $A$ be a matrix and $J$ its Jordan canonical form. How can one express $A$ as a product of two symmetric matrices? I expressed $J$ as a product of two symmetric matrices: block by block in the ...
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### Proof that if a matrix is invertible, its rank is maximum

I have to prove that if a square matrix $A \in \mathfrak{M}_n (\mathbb{K})$ is invertible, then $rg(A) = n$. The thing is I cannot use vector spaces, subspaces, etc... to prove this, only matrix ...
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### Proving that $M_p(M_q (K)) \cong M_{pq} (K)$.

My textbook finishes the proof of one of the theorems with the following fact: $$M_p(M_q (K)) \cong M_{pq} (K),$$ where $K$ is a field, and it says that it is true by "block multiplication ...
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### Name of this matrix product?

Suppose $A$ and $B$ have columns as follows, $$A = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix},$$ $$B = \begin{bmatrix} b_1 & b_2 & \dots & b_n \end{bmatrix}.$$ Is ...
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### Rank of a partitioned matrix

Let $p$, $q$ and $n$ be positive integers such that $p+q \geq n$ and $p \leq n$. Denote by $\mathbf{I}$ the identity matrix of size $p$, by $\mathbf{0}$ the zero matrix of size $(p\times (n-p))$ and ...
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### Proving matrix $A$ is similar to matrix $B$

Question: If the matrix $\begin{pmatrix} A & 0 \\ 0& A \end{pmatrix}$ is similar to $\begin{pmatrix} B & 0 \\ 0 & B \end{pmatrix}$ show that: the matrix $A$ is similar the matrix ...
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I am studying a paper which describes split-radix algorithms by making matrix factorizations, so that e.g. DCT 8x8 can be computed via 4 DCT4x4. I must apologize, but the question is related to an ...
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### Simple to state yet tricky question

Define $$A=\left[\mathrm I+\sum_{k=1}^{m_1}v_k v_k^T+\sum_{k=1}^{m_2}u_k u_k^T\right]^{-1},$$ where each $u_k$ and $v_k$ is a $0$-$1$ column vector, and for each $1\leq i \leq n$, the $i$th component ...
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### Jordan Canonical form 2x2 matrix

Compute the Jordan Canonical form of A = $\begin{bmatrix}i & 1\\1 & -1\end{bmatrix}$. My (feeble) attempt: After I compute the characteristic polynomial, which gives me $x^2=0$, the ...