Linear Algebra - Elementary transformation matrix

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Elementary transformation matrix

Elementary transformation matrix into elementary primary transformation and column transformation
connecting arrows between elementary transformation matrix and the matrix, the equal sign can not be used

Elementary Line

• Exchange two lines
• Multiplying a row by k (k ≠ 0)
• 1 doubly a row to go up a row

Theorem 1
Any matrix can be formed by elementary transformation into a standard (row transformation and column transformation can)

Equivalent : A by elementary transform B, A are called equivalent to B, denoted as

Equivalent nature

Elementary square

Elementary square : unit matrix E made of a matrix of elementary transform is elementary matrix .

1. Elementary inverse square can
2. Its inverse is also elementary square.
3. Elementary matrix transpose is elementary square.

Elementary square :

1. Exchange of i, j row, denoted by E (i, j), the determinant is equal to -1, an inverse matrix E (i, j)
2. Multiplying a row by k (k ≠ 0), denoted by E (i (k)), k ≠ 0, the determinant is equal to k, the inverse matrix E (i (1 / k))
3. L times the j-th row is added to i-th row, denoted by E (i, j (k)), the determinant is equal to 1, the inverse matrix E (i, j (-l))

Theorem 2 : Let A be an arbitrary matrix A by elementary matrix with i-th left (right), equivalent to the embodiment A of the i-th rows (columns) conversion.

Theorem 3 : there are elementary arbitrary matrix A square p1, p1 ··· ps, Q1, Q2, ···, Qt, such that ps, ···, p1AQ1, ···, Qt is the standard form of A.
Corollary : If A, B is equivalent, there is invertible matrix p, Q, such that PAQ = B

Theorem 4 : A necessary and sufficient condition is reversible A standard form of E.
Theorem 5 : A necessary and sufficient condition is reversible A can be expressed as the product of a number of elementary square.

Primary transformation matrix inversion method

Precautions:

1. To the first column, the second column ···, and so on
2. Write the entire line, the entire line operation
3. After processing the first column, the first row is not active transform
4. In doing matrix transform matrix connected by arrows
5. Only primary transformation
6. Whether or not reversible, if the left does not pose identity matrix, then the matrix is ​​irreversible.

Rank of the matrix

A matrix, consisting of any k - k rows and k columns is the k-th order determinant subformulae
rank of the matrix: non-null sub-type of the highest order of a matrix A k is the rank of the matrix , expressed as r (A) = k

For a matrix Am × n, 0 ≤ r (A) ≤ min {m, n}

r (A) = m, taking all the rows, called full row rank
r (A) = n, taking all of the columns, called full rank
if it is full row rank or full rank, we referred to as a full rank

If r (A) <min {m , n}, it is called a reduced rank

If A is a square matrix, A full rank of A necessary and sufficient condition is reversible

Theorem. 1: r (A) = r is a necessary and sufficient condition of a sub-step of formula r is not 0, and all of the r + 1-order sub-formulas are all 0

Stepped:

1. If the zero line, zero line below the zero line
2. From top to bottom, left to right the first non-zero element known as the first non-zero elements, increase the number of accompanying the first nonzero number to the left of zero and strictly increasing

Simplified stepped row *

1. Stepped
2. The first non-zero elements zero line is 1
3. The remaining elements of the column where the first nonzero 0

How to determine whether the simplified line is stepped

1. Videos fold line (determined whether or stepped)
2. Analyzing the first non-zero row of non-zero element is 1
3. Other elements to determine where the first line of non-zero elements of nonzero row is 0

Generally, the number of rows, row echelon form row rank equal to zero

Elementary transformation does not change the rank of the matrix

Example:

Rank nature

Property 1:
Property 2: arbitrary matrix by reversible matrix, his rank invariant
Property 3: The matrix A is a square matrix of n × m, P is the m-order reversible matrix, Q is invertible matrix of order n, R & lt (A ) = r (PA) = r (AQ) = r (PAQ)