Decomposition of Matrices

Find the $LU$-decomposition of the $3 \times 3$ matrix

The triangular matrices $L$ and $U$ are not uniquely determined by the matrix equation $A = LU$. These two matrices together contain $n^{2}+n$ unknown elements. Thus when comparing elements on the left- and right-hand side of $A = LU$ we have $n^{2}$ equations and $n^{2}+n$ unknowns. We require a further $n$ conditions to uniquely determine the matrices. There are three additional sets of $n$ conditions that are commonly used. These are Doolittle’s method with $\ell_{jj} = 1$, $j = 1, \ldots , n$; Choleski’s method with $\ell_{jj} = u_{jj}$, $j = 1, \ldots , n$; Crout’s method with $u_{jj} = 1$, $j = 1, \ldots , n$. Apply Crout's method.

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$. Consider the $LU$-decomposition $A = LU$, where $L$ is a unit lower triangular matrix and $U$ is an upper triangular matrix. The $LDU$-decomposition is defined as $A = LDU$, where $L$ is unit lower triangular, $D$ is diagonal and $U$ is unit upper triangular. Let

Find the $LDU$-decomposition via the $LU$-decomposition.

Find the $QR$-decomposition of the $3 \times 3$ matrix

We consider $3 \times 3$ matrices over $\mathbb {R}$. An orthogonal matrix $Q$ such that $\operatorname {det}\left(Q\right) = 1$ is called a rotation matrix. Let $1 \leq p < r \leq 3$ and $\phi$ be a real number. An orthogonal $3 \times 3$ matrix $Q_{pr}(\phi ) = (q_{ij})_{1 \leq i, j \leq 3}$ given by

will be called a plane rotation through $\phi$ in the plane span $(e_{p}, e_{r})$. Let $Q = (q_{ij})_{1 \leq i, j \leq 3}$ be a rotation matrix. Show that there exist angles $\phi \in [0, \pi )$, $\theta , \psi \in (-\pi , \pi ]$ called the Euler angles of $Q$ such that

Consider a square non-singular square matrix $A$ over $\mathbb {C}$, i.e. $A^{-1}$ exists. The polar decomposition theorem states that $A$ can be written as $A = UH$, where $U$ is a unitary matrix and $H$ is a hermitian positive definite matrix. Show that $A$ has a unique polar decomposition.

For any $n \times n$ matrix $A$ over $\mathbb {C}$, there exists a positive semi-definite matrix $H$ and a unitary matrix such that $A = HU$. If $A$ is nonsingular, then $H$ is positive definite and $U$ and $H$ are unique.

Let $A$ be an arbitrary $m \times n$ matrix over $\mathbb {R}$, i.e. $A \in \mathbb {R}^{m \times n}$. Then $A$ can be written as

where $U$ is an $m \times m$ orthogonal matrix, $V$ is an $n \times n$ orthogonal matrix, $\Sigma$ is an $m \times n$ diagonal matrix with nonnegative entries and the superscript $T$ denotes the transpose. This is called the singular value decomposition. An algorithm to find the singular value decomposition is as follows.

1. Find the eigenvalues $\lambda_{j}$ ($j = 1, \ldots , n$) of the $n \times n$ matrix $A^{T}A$. Arrange the eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ in descending order.

2. Find the number of nonzero eigenvalues of the matrix $A^{T}A$. We call this number $r$.

3. Find the orthogonal eigenvectors $\overrightarrow {v}_{j}$ of the matrix $A^{T}A$ corresponding to the obtained eigenvalues, and arrange them in the same order to form the column-vectors of the $n \times n$ matrix $V$.

4. Form an $m \times n$ diagonal matrix $\Sigma$ placing on the leading diagonal of it the square root $\sigma_{j} := \sqrt{\lambda_{j}}$ of $p = \min (m, n)$ first eigenvalues of the matrix $A^{T}A$ found in 1) in descending order.

5. Find the first $r$ column vectors of the $m \times m$ matrix $U$

6. Add to the matrix $U$ the rest of the $m-r$ vectors using the Gram-Schmidt orthogonalization process.

We have $A \overrightarrow {v}_{j} = \sigma_{j} \overrightarrow {u}_{j}$, $A^{T} \overrightarrow {u}_{j} = \sigma_{j} \overrightarrow {v}_{j}$ and therefore $A^{T}A \overrightarrow {v}_{j} = \sigma_{j}^{2} \overrightarrow {v}_{j}$, $AA^{T} \overrightarrow {u}_{j} = \sigma_{j}^{2} \overrightarrow {u}_{j}$. Apply the algorithm to the matrix

Find the singular value decomposition $A = U \Sigma V^{T}$ of the matrix (row vector) $A = (2 \ 1 \ -2)$.

Let $A$ be an $n \times n$ matrix over $\mathbb {R}$. Assume that $A^{-1}$ exists. Given the singular value decomposition of $A$, i.e. $A = UWV^{T}$. Find the singular value decomposition for $A^{-1}$.

Find the Moore-Penrose pseudo inverses of

applying the singular value decomposition.

Any unitary $2^{n} \times 2^{n}$ matrix $U$ can be decomposed as

where $U_{1}, U_{2}, U_{3}, U_{4}$ are $2^{n-1} \times 2^{n-1}$ unitary matrices and $C$ and $S$ are the $2^{n-1} \times 2^{n-1}$ diagonal matrices

where $\alpha_{j} \in \mathbb {R}$. This decomposition is called cosine-sine decomposition.

Find the cosine-sine decomposition of the $4 \times 4$ unitary matrix (Bell matrix)

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. Then there exists an $n \times n$ unitary matrix $(U^{*} = U^{-1})$ such that

where $T$ is an $n \times n$ matrix in upper triangular form. Equation (1) is called a Schur decomposition. The diagonal elements of $T$ are the eigenvalues of $A$. Note that such a decomposition is not unique. An iterative algorithm to find a Schur decomposition for an $n \times n$ matrix is as follows.

It generates at each step matrices $U_{k}$ and $T_{k}$ ($k = 1, \ldots , n-1$) with the properties: each $U_{k}$ is unitary, and each $T_{k}$ has only zeros below its main diagonal in its first $k$ columns. $T_{n-1}$ is in upper triangular form, and $U = U_{1}U_{2} \cdots U_{n-1}$ is the unitary matrix that transforms $A$ into $T_{n-1}$. We set $T_{0} = A$. The $k$-th step in the iteration is as follows.

Step 1. Denote as $A_{k}$ the $(n-k+1) \times (n-k+1)$ submatrix in the lower right portion of $T_{k-1}$.

Step 2. Determine an eigenvalue and the corresponding normalized eigenvector for $A_{k}$.

Step 3. Construct a unitary matrix $N_{k}$ which has as its first column the normalized eigenvector found in step 2.

Step 4. For $k = 1$, set $U_{1} = N_{1}$, for $k > 1$, set

where $I_{k-1}$ is the $(k-1) \times (k-1)$ identity matrix.

Step 5. Calculate $T_{k} = U_{k}^{*}T_{k-1}U_{k}$.

Apply the algorithm to the $3 \times 3$ symmetric matrix

Let $A$ be an $n \times n$ matrix over $\mathbb {C}$. Then there exists an $n \times n$ unitary matrix $Q$, such that

where $D = \operatorname {diag}(\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n})$ is the diagonal matrix composed of the eigenvalues of $A$ and $N$ is a strictly upper triangular matrix (i.e. $N$ has zero entries on the diagonal). The matrix $Q$ is said to provide a Schur decomposition of $A$. Let

Show that $Q$ provides a Schur decomposition of $A$.

We say that a matrix is upper triangular if all their entries below the main diagonal are 0, and that it is strictly upper triangular if in addition all the entries on the main diagonal are equal to 1. Any invertible real $n \times n$ matrix $A$ can be written as the product of three real $n \times n$ matrices

where $N$ is strictly upper triangular, $D$ is diagonal with positive entries, and $O$ is orthogonal. This is known as the Iwasawa decomposition of the matrix $A$. The decomposition is unique. In other words, that if $A = O'D'N'$, where $O'$, $D'$ and $N'$ are orthogonal, diagonal with positive entries and strictly upper triangular, respectively, then $O' = O$, $D = D'$ and $N' = N$.

Let $A$ be a unitary $n \times n$ matrix. Let $P$ be an invertible $n \times n$ matrix. Let $B := AP$. Show that $PB^{-1}$ is unitary.

Show that every $2 \times 2$ matrix $A$ of determinant 1 is the product of three elementary matrices. This means that matrix $A$ can be written as

Almost any $2 \times 2$ matrix $A$ can be factored (Gaussian decomposition) as

Find the decomposition of the matrix

Consider the symmetric $3 \times 3$ matrix $A$ over $\mathbb {R}$ and the orthogonal matrix $O$, respectively

Calculate $\widetilde{A} = O^{-1}AO$. Can we find an angle $\phi$ such that $\widetilde{a}_{12} = \widetilde{a}_{21} = 0$?

Let $A$ be an $m \times m$ matrix. Let $B$ be an $n \times n$ matrix. Let $X$ be an $m \times n$ matrix such that

We can find non-singular matrices $V$ and $W$ such that $V^{-1}AV = J_{A}$, $W^{-1}BW = J_{B}$, where $J_{A}$, $J_{B}$ are the Jordan canonical form of $A$ and $B$, respectively. Show that from (1) it follows that $J_{A}Y = YJ_{B}$, where $Y := V^{-1}XW$.

Write the matrix

as the product of two $4 \times 4$ matrices $A$ and $B$ such that each of these matrices has precisely two non-zero entries in each row.

An $n \times n$ matrix $A$ is persymmetric if $J_{n}AJ_{n} = A^{T}$ and skew persymmetric if $J_{n}AJ_{n} = A^{T}$, where $J_{n}$ is the $n \times n$ exchange matrix, i.e. the matrix with 1's on the counter diagonal and 0's otherwise. Hence $J_{n}^{2} = I_{n}$. Let $B$ be an $n \times n$ symmetric matrix.

Decompose the $3 \times 3$ matrix

as the sum of a persymmetric and a skew persymmetric matrix.

If $A \in \mathbb {R}^{n \times n}$, then there exists an orthogonal $Q \in \mathbb {R}^{n \times n}$ such that

where each $R_{ii}$ is either a $1 \times 1$ matrix or a $2 \times 2$ matrix having complex conjugate eigenvalues. Find $Q$ for the $3 \times 3$ matrix

Then calculate $Q^{T}AQ$.

Find a cosine-sine decomposition of the matrices

The Cholesky decomposition of the a positive semi-definite matrix $M$ is the decomposition into a product of a lower triangular matrix $L$ and its conjugate transpose $L^{*}$, i.e. $M = LL^{*}$. Find the Cholesky decomposition for the density matrix (pure state)

Show that (cosine-sine decomposition)

Let $U$ be an $n \times n$ unitary matrix. The matrix $U$ can always be diagonalized by a unitary matrix $V$ such that

where $e^{i\theta_{j}}$, $\theta_{j} \in [0, 2\pi )$ are the eigenvalues of $U$. Let $n = 2$ and

Thus the eigenvalues of $U$ are 1 and $-1$. Show that

If $A \in SL(2, \mathbb {R})$, then it can be uniquely be written in the form

Find this decomposition for the $2 \times 2$ nonnormal matrix