Finding Contract Transformation of Block Matrix by Elementary Transformation Method

1. Use the "elementary transformation method" to find the diagonal matrix contracted by the symmetric matrix

In order to make this article complete, here is how to use the "elementary transformation method" to find the diagonal matrix contracted by the symmetric matrix.

When learning linear algebra, we know that for any $n\times$Real symmetric matrixAA$of\; n$$A$ , you can use the "elementary transformation method" to find the invertible matrix$C$ and Diagonal Matrix$D$ , such that$A$ and$D$ contract, ie$C^{T}AC=D$ , where$C^T$ stands forCCTranspose of$C.$

Specific method: make $2n\times n-$ matrix

$$\left[\begin{array}{c}A\\ I\end{array}\right]\underset{Apply\; the\; same\; elementary\; column}{\overset{Perform\; elementary\; row\; transformationonA}{\to }}\left[\begin{array}{c}D\\ C\end{array}\right]$$

则 $C^{T}AC=D$。

Note that an elementary row transformation needs to be applied first, and then the same elementary column transformation is applied to the entire column at once.

Example 1

Known real symmetric matrix
$$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 5\end{array}\right]$$
Finding the Reversible Matrix CC by Elementary Transformation Method$C$ and diagonal matrix$D$ , such that$A$ and$D$ contract.
$$\left[\begin{array}{c}A\\ I\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 5\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\underset{c_2-c_1}{\overset{r_2-r_1}{\to }}\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\\ 1& 2& 5\\ 1& -1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\underset{c_3-c_1}{\overset{r_3-r_1}{\to }}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 2\\ 0& 2& 4\\ 1& -1& -1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\underset{c_3-c_2}{\overset{r_3-r_2}{\to }}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\\ 1& -1& 1\\ 0& 1& -2\\ 0& 0& 1\end{array}\right]$$

The required invertible matrix $C$ and diagonal matrix$D$ also
$$C=\left[\begin{array}{ccc}1& -1& 1\\ 0& 1& -2\\ 0& 0& 1\end{array}\right],D=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$$

且 $C^{T}AC=D$。

2. Use the "elementary transformation method" to find the block diagonal matrix contracted by the block symmetric matrix

Next, we extend the real symmetric matrix to the block symmetric matrix, and the elements in the matrix are no longer limited to real numbers, but can be complex numbers. By analogy to the above elementary transformation method without proof, we give the elementary transformation method for finding the block diagonal matrix contracted by the block symmetric matrix.

For an $n\times$Block Symmetric MatrixAA$of\; n$$A$ , you can use the "elementary transformation method" to find the invertible matrix$C$ and Block Diagonal Matrix$D$ , such that$A$ and$D$ contract, namely$C^{H}AC=D$ , where$C^H$ meansCCThe conjugate transpose of$C.$

Specific method: make $2n\times$Matrix of$n$
$$\left[\begin{array}{c}A\\ I\end{array}\right]\underset{Apply\; the\; same\; elementary\; column}{\overset{Perform\; elementary\; row\; transformationonA}{\to }}\left[\begin{array}{c}D\\ C\end{array}\right]$$
则 $C^{H}AC=D$。

In addition, it is also necessary to understand the elementary transformation rules of the block matrix, as follows:

• Elementary Row Transformation Rules for Block Matrix

  • Put a block row to the left $P$ times ($P$ is the matrix) added to another block row;
  • Swap the position of two block lines
  • Left -multiply a block of rows by an invertible matrix

• Elementary Column Transformation Rules of Block Matrix

  • Put a block column on the right $P$ times ($P$ is matrix) added to another block column;
  • Swap the positions of two block columns
  • Right- multiply a block of columns by an invertible matrix

• It should be noted that there is a difference between "left" and "right" in row transformation and column transformation.

• And (emphasis added), if you put a block row on the left $P$ times ($P$ is a matrix) is added to another block row, and the corresponding elementary column transformation is to convert theright $P^H$ (note the conjugate transpose here) is multiplied to another block column.

It will be easier to understand directly by looking at the example below.

Example 2

Environmental value
$$B=\left[\begin{array}{cc}A& a\\ a^H& \frac{1}{l_0}\end{array}\right]$$
Among them$A$ is$n\times The\; Hermite\; matrix\; of\; n$ (ie$A^H=A$ ), and$A$ is positive definite,$\alpha$ is$n-$ dimensional unit column vector. Finding the Reversible Matrix CCby Elementary Transformation Method$C$ and diagonal matrix$D$ , such that$B$ and$D$ contract.

$$\left[\begin{array}{c}B\\ I\end{array}\right]=\left[\begin{array}{cc}A& a\\ a^H& \frac{1}{l_0}\\ I_n& O\\ O& 1\end{array}\right]\underset{c_2-c_1(a^H{A}^{-1}{)}^H}{\overset{r_2-a^H{A}^{-1}{r}_1}{\to }}\left[\begin{array}{cc}A& O\\ O& \frac{1}{l_0}-a^H{A}^{-1}a\\ I_n& -(A^H{)}^{-1}a\\ O& 1\end{array}\right]$$

The required invertible matrix $C$ and diagonal matrix$D$ are

$$C=\left[\begin{array}{cc}{I}_n& -(A^H{)}^{-1}a\\ O& 1\end{array}\right],D=\left[\begin{array}{cc}A& O\\ O& \frac{1}{l_0}-a^H{A}^{-1}\end{array}\right]$$
且 $C^{H}AC=D$。

It should be noted that because $B$ is a block matrix, so the identity matrix$I$ also score blocks, and the block method should be the same as that$B$ agrees.

3. Positive definiteness of application-judgment matrix

Look at an engineering matrix (may also come from advanced algebra) topic:

Find a block diagonal matrix, which is consistent with the original block symmetric matrix. It is often used to analyze the positive definiteness of a block matrix, such as the following question.
Answer:

Use EE in the answer to the above question$E$ represents the unit matrix, and IIis used in the previous article$I$ need to pay attention.

The first step in this question is to find the matrix of the conjugate contract with the original matrix, which is to use the method of this article.
In addition, if the two matrices are conjugated, their positive definiteness will remain consistent, so a new matrix that is conjugated to the original matrix can be obtained through elementary transformations, and the positive definiteness of this new matrix is ​​easy to prove, then the original The positive definiteness of the matrix is ​​then proved.