4. Determinants - Yonsei · 2014. 12. 29. · 4.1. Determinants; Cofactor Expansion Cofactor...

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4. Determinants4. Determinants


4.1. Determinants; Cofactor Expansion4.1. Determinants; Cofactor Expansion

Determinants of 2×2 and 3×3 Matrices

2×2 determinant



Determinants of 2×2 and 3×3 Matrices

3×3 determinant



Elementary Products

(4)

Elementary product: a product containing one entry from each row and one entry from each column.

In formula (4) each elementary product is of the form

where the blank contain some permutation of the column indices {1, 2, 3}.

The sign can be determined by counting the minimum number of interchanges in the permutation of the column indices required to put those indices into their natural order: the sign is + if the number is even and – if it is odd.

Signed elementary product



Elementary Products



General Determinants

n×n determinant or n-th order determinant

The signed elementary products are to be summed over all possible permutations {j1, j2, …, jn} of the column indices.



Determinants of Matrices with Rows or Columns That Have All Zeros

Determinants of Triangular Matrices



Minors and Cofactors



Minors and Cofactors

Example 3Example 3



Cofactor Expansions

These are called cofactor expansions of A. Note that in each cofactor expansion, the entries and cofactors all come from the same row or the same column.



Cofactor Expansions



Cofactor Expansions

Example 5Example 5Use a cofactor expansion to find the determinant of


4.2. Properties of Determinants4.2. Properties of Determinants

Determinant of AT



Effect of Elementary Row Operations on A Determinant




Suppose that the ith row of A is multiplied by the scalar k to produce the matrix B,

Since the ith row is deleted when the cofactors along that row are computed, the cofactors in this formula are unchanged when the ith row is multiplied by k.

(a)




Example 1Example 1




Example 2Example 2




Example 3Example 3What is the relationship between det(A) and det(-A)?

Thus, det(-A)=det(A) if n is even, and det(-A)=-det(A) if n is odd.



Simplifying Cofactor Expansions

(1) A cofactor expansion can be minimized by expanding along a row or column with the maximum number of zeros.

(2) Adding multiples of one row (or column) to another does not change the determinant of the matrix.

Example 4Example 4

Use a cofactor expansion to find the determinant of



A Determinant Test for Invertibility

If R is the reduced row echelon form of square matrix A, determinant of the matrices are both zero or both nonzero.

Assume that A is invertible, in which case the reduced row echelon form of A is I (by Theorem 3.3.3). Since det(I)≠0, it follows that det(A)≠0.

Conversely, assume that det(A)≠0. Since det(R)≠0, R=I. Thus, A is invertible (by Theorem 3.3.3)



Determinants of A Product of Matrices

Determinants of The Inverse of A Matrix

Since AA-1=I, det(AA-1)=det(I)=1=det(A)det(A-1). Thus, det(A-1)=1/det(A).



Determinants of A+B

det(A+B)≠det(A)+det(B)

Example 8Example 8

det(A)=1, det(B)=5, det(A+B)=23.



Determinant Evaluation by LU-Decomposition

If A=LU, then det(A)=det(L)det(U), which is easy to compute since L and U are triangular.

Thus, nearly all of the computational work in evaluating det(A) is expended in obtaining the LU-decomposition.

From Table 3.7.1.The number of flops required to obtain the LU-decomposition of an n×n matrix is on the order of 2/3n3 for large values of n.This is an enormous improvement over the determinant definition, which involves the computation of n! signed elementary products.

For example, today’s typical PC can evaluate 30×30 determinant in less than one-thousandth of a second by LU-decomposition compared to the roughly 1010 years that would be required for it to evaluate 30! Signed elementary products.



A Unifying Theorem


4.3. Cramer4.3. Cramer’’s Rules Rule

Adjoint of A Matrix



Adjoint of A Matrix

If we multiply the entries in the first row by the corresponding cofactors from the third row, then the sum is

Let’s consider a matrix A’ that results when the third row of A is replaced by a duplicate of the first row.

We know that det(A’)=0 because of the duplicate rows.



Adjoint of A Matrix



Adjoint of A Matrix

Example 1Example 1Find the matrix of cofactors from A.



A Fomula for The Inverse of A Matrix

Suppose that A is invertible.

The entry in the ith row and jth column of this product is

(3)




In the case where i=j, the entries and cofactors come from the same row of A, so (3) is the cofactor expansion of det(A) along that row.

In the case where i≠j, the entries and cofactors come from different rows, so the sum is zero by Theorem 4.3.1.

Since A is invertible, it follows that det(A)≠0, so this equation can be rewritten as




Example 2Example 2Find the inverse of the matrix A in Example 1.



How The Inverse Formula Is Used

Computer program usually use LU-decomposition (as discussed in Section 3.7) and not Formula (2) to invert matrices.

Thus, the value of Formula (2) is not for numerical computations, but rather as a tool in theoretical analysis.



Cramer’s Rule



Cramer’s Rule

In the case where det(A)≠0, we can use Formula (2) to rewrite the unique solution of Ax=b as

Therefore, the entry in the jth row of x is

The cofactors in this expression come from the jth column of A and hence remain unchanged if we replace the jth column of A by b (the jth column is crossed out when the cofactors are computed). Since this substitution produces the matrix Aj, the numerator in (5) can be interpreted as the cofactor expansion along the jth column of Aj. Thus,



Cramer’s Rule

Example 4Example 4Use Cramer’s rule to solve the system



Cramer’s Rule

Example 5Example 5Use Cramer’s rule to solve the system



Geometric Interpretation of Determinants




(a)

Suppose that the matrix A is partitioned into columns as

and let us assume that the parallelogram with adjacent sides u and v is not degenerate.

We can see that this area can be expressed as




The square of the area can be expressed as

(area)2=




Example 8Example 8Find the area of the triangle with vertices A(-5, 4), B(3, 2), and C(-2, -3).



Cross Products

Standard unit vector i=(1,0,0), j=(0,1,0), k=(0,0,1)



Cross Products

Example 9Example 9Let u=(1,2,-2) and v=(3,0,1). Find

(a) u×v (b) v×u (c) u×u



Cross Products



Cross Products

In general, if u and v are nonzero and nonparallel vectors, then the direction of u×v in relation to u and v can be determined by the right-hand rule.



Cross Products

The associative law does not hold for cross products; for example,



Cross Products



Cross Products

(a) Since 0≤ θ ≤ π, it follows that sin θ ≥0 and hence that

Thus,



Cross Products

(b)

Example 10Example 10

Find the area of the triangle in R3 that has vertices P1(2,2,0), P2(-1,0,2), and P3(0,4,3).


4.4. A First Look at Eigenvalues and Eigenvectors4.4. A First Look at Eigenvalues and Eigenvectors

Fixed Points

Recall that a fixed point of an n×n matrix A is a vector x in Rn such that Ax=x (see the discussion preceding Example 6 of Section 3.6)

Every square matrix A has at least one fixed point, namely x=0. We call this the trivial fixed point of A.

Ax=x Ax=Ix (A-I)x=0



Eigenvalues and Eigenvectors

One might also consider more general equations of the form Ax= λx in which λ is a scalar.




The most direct way of finding the eigenvalues of an n×n matrix A is to rewrite the equation Ax= λx as Ax= λIx, or equivalently, as

and then try to determine those values of λ, if any, for which this system has nontrivial solutions. Since (4) has nontrivial solutions if and only if the coefficient matrix λI-A is singular, we see that the eigenvalues of A are the solutions of the equation

(4)

This is called the characteristic equation of A. Also, if λ is an eigenvalue of A, then (4) has a nonzero solution space, which we call the eigenspace of A corresponding to λ.

It is the nonzero vectors in the eigenspace of A corresponding to λ that are the eigenvectors of A corresponding to λ.




Example 2Example 2(a) Find the eigenvalues and corresponding eigenvectors of the matrix

(b) Graph the eigenspaces of A in an xy-coordiante system.



Eigenvalues of Triangular Matrices

If A is an n×n triangular matrix with diagonal entries a11, a22, …, ann, then λI-A is a triangular matrix with diagonal entries λ-a11, λ-a22, …, λ-ann. Thus the characteristic polynomial of A is

which implies that the eigenvalues of A are



Eigenvalues of Powers of A Matrix

If λ is an eigenvalue of A and x is a corresponding eigenvector, then

which shows that λ2 is an eigenvalue of A2 and x is a corresponding eigenvector.



A Unifying Theorem

λ=0 is an eigenvalue of A if and only if there is a nonzero vector x such that Ax=0.



Algebraic Multiplicity

REMARK

This theorem implies that an n×n matrix has n eigenvalues if we agree to count repetitions and allow complex eigenvalues, but the number of distinct eigenvalues may be less than n.



Eigenvalue Analysis of 2×2 Matrices



Eigenvalue Analysis of 2×2 Symmetric Matrices

Later in the text we will show that all symmetric matrices with real entries have real eigenvalues.




If the 2×2 symmetric matrix is

then

so Theorem 4.4.9 implies that A has real eigenvalues.

It also follows from that theorem that A has one repeated eigenvalue if and only if

Since this holds if and only if a=d and b=0.




Example 6Example 6Graph the eigenspaces of the symmetric matrix

in an xy-coordinate system.



Expressions for Determinant and Trace in Terms of Eigenvalues

(a)

Write the characteristic polynomial in factored form:

Setting λ=0 yields

But det(-A)=(-1)ndet(A), so it follows that




(b)

Assume that A=[aij], so we can write p(λ) as

Any elementary product that contains an entry that is off the main diagonal of (30) as a factor will contain at most n-2 factors that involve λ. Thus the coefficient of λn-1 in p(λ) is the same as the coefficient of λn-1 in the product

(30)

Expanding this product we see that it has the form




and expanding the expression for p(λ) we see that it has the form

=

Thus, we must have




Example 7Example 7Find the determinant and trace of a 3×3 matrix whose characteristic polynomial is

This polynomial can be factored as



Eigenvalues by Numerical Methods

Eigenvalues are rarely obtained by solving the characteristic equation in real-world applications primarily for two reasons:

1. In order to construct the characteristic equation of an n×n matrix A, it is necessary to expand the determinant det(λI-A). The computations are prohibitive for matrices of the size that occur in typical applications.

2. There is no algebraic formula of finite algorithm that can be used to obtain the exact solutions of the characteristic equation of a general n×n matrix when n≥5.

Given these impediments, various algorithms have been developed for producing numerical approximations to the eigenvalues and eigenvectors.

4. Determinants - Yonsei · 2014. 12. 29. · 4.1. Determinants; Cofactor Expansion Cofactor...

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