# Questions tagged [diagonalization]

For questions about matrix diagonalization, that is, writing a matrix, a bilinear form or an operator into a "basis" making this one diagonal. This tag is **NOT** for diagonalization arguments from logic and set theory.

1,462 questions
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### Jordan normal as transformation with respect to the basis of eigenvectors

I have the follwing matrix A: A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 ...
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### Prove that all normal matrices are semi-simple using Schur's decomposition

Is there any elegant proof that shows that all normal matrices are semi-simple that comes from Schur's decomposition or its corrolaries? There is a proof that normal matrices are unitary diagonizable ...
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### Diagonizability of $n \times n$ matrix with two distinct eigenvalues and eigenspace dimension $n-1$

Let $\mathbf{A}$ be an $n \times n$ matrix with two distinct eigenvalues $\lambda_1$ and $\lambda_2$. If the dimension of the eigenspace $E(\lambda_1)$ is $n-1$, then $\mathbf{A}$ is diagonizable. ...
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### Why not different diagonal elements in orthogonal reduction of quadratic form?

Suppose we have quadratic form $(3x)^2 -(2y)^2-z^2-(4xy)+(8xz)+(12yz)$ and we are asked to reduce it to cannonical form. The steps would be: Find eigen values (3,6,-9 in this case) of the matrix of ...
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### Knowing $M^2 + M = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, find the eigenvalues of $M$

Let $M \in M_2(\mathbb R)$ and $$A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$ such that $$M^2 + M = A$$ Find the eigenvalues of $M$. Here is what I did: Let $\lambda$ be an eigenvalue ...
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### Generalised Eigenvectors Issue

Ok so i have been doing a few questions on 'Diagonalising' defective matrices, the method I've been using to find generalized Eigenvectors is to make the previous Eigenvector the subject. However i ...
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### If $A^2 + b A + c I = 0$ why does $A$ have to be diagonalizable?

If $A \in \mathbb C^{n \times n}$, $b, c$ are scalars such that $c \ne \frac 14 b^2$ and $$A^2 + b A + c I = 0$$ why does $A$ have to be diagonalizable? I am getting to a point where I think $A$ has ...
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### Show that a matrix is diagonalizable for all values of a

I have to show that the matrix A is Diagonalizable for all values of a. Given the matrix A. \begin{pmatrix} a+3 & 4 \\ 5 & 5 \end{pmatrix} I first found the characteristic polynomial for A:...
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### why any matrix of $SU(2)$ is diagonalizable?

"Any matrix $A\in SU(2)$ can be written as $\begin{pmatrix}z & -\overline w \\ w & \overline z\end{pmatrix}$ for some $w,z\in\mathbb C$ with $|w|^2+|z|^2=1$. Moreover, every matrix of this ...
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### Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$, for what values of $a$ given matrix must be diagonalizable.

Let $A$ be a $3\times 3$ matrix with characteristic polynomial $x^3-3x+a$. For what values of $a$ given matrix must be diagonalizable. I am talking about diagonalizability over reals. Efforts: If a ...
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### What value of b make the following matrix diagonalisable

$$\begin{bmatrix} 1 & -1 \\ 1 & b \\ \end{bmatrix}$$ It seems like there's no two distinct eigenvalues to make it diagonalisable. Can anyone confirm that is true?
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### Proving that any unitary matrix can be diagonalised by a similar matrix

I'm having struggles with understanding important facts about spectral theorem in finite dimensional spaces. For hermitian matrices, I saw in classes that the similarity matrix that diagonalises any ...
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### Existence of diagonalizable matrix

Let $A \in \mathbb{C}^{4 \times4}$ and $A^5 = I$. Is $A$ always diagonalizable? How to show $|\mbox{tr} (A)| \leq 4$? If $\mbox{tr} (A) = 4$, can we determine matrix $A$? Some ...
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### How can I block diagonalise this matrix?

I have this matrix: $$A = \left( \begin{array}{cccc} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ \end{array} \right)$$ ...
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### Representations of $\mathbb{G}_m$

I know that the multiplicative affine group scheme $\mathbb{G}_m$ is diagonalizable, since the algebra that represents it is $k[X,X^{-1}]$, which is isomorphic to the group algebra $k[\mathbb{Z}]$. ...
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### How come a matrix not diagonalizable if the geometric multiplicity is less than the algebraic multiplicity?

I understand that the algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial. I also understand that the geometric multiplicity is the dimension ...
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### If zero is an eigenvalue are dimensions lost?

This is likely a silly question so sorry in advance. However, I am wondering if I am right in thinking that if zero is an eigenvalue, then some dimension must be lost. My understanding is that ...
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### Similarity between a matrix and diagonal matrix

How to prove a matrix and a diagonal matrix are similar? Is there some rule to follow or are there some steps to follow? Like: suppose a matrix $A$ = something, prove that $A$ is similar to a ...
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### Properties of $n \times n$ complex matrix with $A^m = I$

If $A^m = I_n$ then what can we say about the eigenvalues and diagonalizablity of $A$? The equation given above is an annihilating polynomial of $A$ and therefore minimal polynomial divides it. Since ...
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### Are all matrices almost diagonalizable?

Every real $2$ by $2$ matrix that is not diagonalizable is similar to the $2$ by $2$ jordan canonical form, $$J_2=\begin{bmatrix}s&1\\0&s\end{bmatrix},$$ where $s$ is the eigenvalue (with ...
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### (Unitary) diagonalization of $A = I-xy^*$

I'm continuing to prepare for a Linear Algebra exam and found another problem that puzzles me. Let $A = I+xy^*$, where $x,y \in \mathbb{C}^m (\neq 0)$. (a) Determine a necessary and ...
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### Alternative definition of diagonalisable transformation

Supposedly a transformation $T: V \to V$ is diagonalisable iff there exists a basis of $V$ consisting only of eigenvectors of $T$. Can someone show me why this is true? I don't really know where to ...
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### Eigenvalues, diagonalization and convergence of matrices

I am trying to wrap my head around some basic results in Linear Algebra. I am trying to avoid more abstract concepts like Rank-Nullity, and stay in simple properties at an introductory level. (I've ...
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### Block-diagonalizing an antisymmetric matrix

I was wondering how to block-diagonalize a $10 \times 10$ antisymmetric matrix into block matrices along the diagonal. Can I just diagonalize each non-diagonal block? Thanks!
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### Is a Jordan block not further block diagonalizable?

We can always find a Jordan canonical form of a matrix A. It is a block diagonal matrix. Is it true that each block cannot be reduced to a matrix with more blocks in diagonal? In other words, for a ...
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### When can two matrices have zero diagonal in the same basis?

It's common to ask if two Hermitian matrices $A$ and $B$ can be diagonalized in the same basis. Is there an efficient way to check if they can be made hollow in the same basis? By hollow, I mean that ...
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### If $A$ is in $\mathbb R^{n \times n}$, then is $A=-A^*$ diagonalizable? [closed]

If $A$ is in $\mathbb R^{n \times n}$, then is $A=-A^*$ diagonalizable?
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### show that $C(C[u])=K[u]$

Let $E$ be a $K$-vector space of dimension $n$, and $u$ be a diagonalizable linear map from $E$ to $E$. Let $C[u]$ be the set of linear maps from $E$ to $E$ which commute with $u$. In other words,...
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### What would be the restrictions to diagonalization of this type transformation of Laplacian matrix?

This question is a little specific. I have read about graph theory and saw in one place the following transformation of an Laplacian matrix $L$: \begin{eqnarray} ULW=\begin{bmatrix} -(l_{11}-l_{22})&...
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### Find 3x3 matrix by determinant and 2 eigenvalues/-vectors

I have two eigenvectors: $(2, 1, -1)'$ with eigenvalue $1$, and $(0, 1, 1)'$ with eigenvalue $2$. The corresponding determinant is $8$. How can I calculate the $3\times3$ symmetric matrix $A$ and $AP$?...
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### Is $A \in \mathcal{M}_n(\mathbb{C})$ diagonalizable?

$A \in \mathcal{M}_n(\mathbb{C})$ such that $A^2$ has got $n$ distinct non zero eigenvalues. Show that A is diagonalizable. Attempt : As $A^2$ has got $n$ distinct non zero eigenvalues. The ...
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### Showing that two matrices are similar by showing they are similar to the same diagonal matrix

The exercise gives two matrices $A$ and $B$ and asks you to show they are similar by showing that they are similar to the same diagonal matrix, and then after that find and invertible matrix $P$ such ...
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### Can you tell how many eigenvectors a matrix has from just the characteristic equation?

If the equation has a repeated root, can you tell without evaluating in the matrix if that repeated root corresponds to more than one eigenvector?
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### When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
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### Diagonalizing a real normal matrix

Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{pmatrix}$, how would I find a real orthogonal matrix $P$ such that $PAP^t$ is a diagonal matrix? ...