Eigenvalues and Eigenvectors
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1 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION
2 DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling), and let r be the number of such row interchanges. Then the determinant of A, written as det A, is ( 1) r times the product of the diagonal entries u 11,, u nn in U. If A is invertible, then u 11,, u nn are all pivots (because A I n and the u ii have not been scaled to 1s). Slide 5.2-2
3 DETERMINANATS Otherwise, at least u nn is zero, and the product u 11 u nn is zero. Thus det product of r ( 1) A = pivots in U 0,, when A is invertible when A is not invertible Slide 5.2-3
4 DETERMINANATS A = Example 1: Compute det A for. Solution: The following row reduction uses one row interchange: A = U Slide 5.2-4
5 DETERMINANATS 1 So det A equals. The following alternative row reduction avoids the row interchange and produces a different echelon form. The last step adds times row 2 to row 3: A This time det A is as before. ( 1) (1)( 2)( 1) = 2 1/ = U / 3 0 ( 1) (1)( 6)(1/ 3) = 2 2, the same Slide 5.2-5
6 THE INVERTIBLE MATRIX THEOREM (CONTINUED) n n Theorem: Let A be an matrix. Then A is invertible if and only if: s. The number 0 is not an eigenvalue of A. t. The determinant of A is not zero. Theorem 3: Properties of Determinants n n Let A and B be matrices. a. A is invertible if and only if det. det AB = (det A)(det B) det A = det A b.. T c.. A 0 Slide 5.2-6
7 PROPERTIES OF DETERMINANTS d. If A is triangular, then det A is the product of the entries on the main diagonal of A. e. A row replacement operation on A does not change the determinant. A row interchange changes the sign of the determinant. A row scaling also scales the determinant by the same scalar factor. Slide 5.2-7
8 THE CHARACTERISTIC EQUATION Theorem 3(a) shows how to determine when a matrix of the form is not invertible. A I det( A I) = 0 The scalar equation characteristic equation of A. is called the n n A scalar is an eigenvalue of an matrix A if and only if satisfies the characteristic equation det( A I) = 0 Slide 5.2-8
9 THE CHARACTERISTIC EQUATION Example 2: Find the characteristic equation of A = Solution: Form A I, and use Theorem 3(d): Slide 5.2-9
10 THE CHARACTERISTIC EQUATION The characteristic equation is or det( A I) = det = (5 )(3 )(5 )(1 ) 2 (5 ) (3 )(1 ) = 0 2 ( 5) ( 3)( 1) = 0 Slide
11 THE CHARACTERISTIC EQUATION Expanding the product, we can also write = 0 n n det( A I) If A is an matrix, then is a polynomial of degree n called the characteristic polynomial of A. The eigenvalue 5 in Example 2 is said to have multiplicity 2 because ( 5) occurs two times as a factor of the characteristic polynomial. In general, the (algebraic) multiplicity of an eigenvalue is its multiplicity as a root of the characteristic equation. Slide
12 SIMILARITY n n If A and B are matrices, then A is similar to B if 1 there is an invertible matrix P such that P AP = B, or, equivalently,. A = PBP 1 Q BQ 1 P 1 Writing Q for, we have. = A So B is also similar to A, and we say simply that A and B are similar. Changing A into transformation. 1 P AP is called a similarity Slide
13 SIMILARITY Theorem 4: If matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities). Proof: If B = n n 1 P AP then, B I = P AP P P = P ( AP P) = P ( A I) P Using the multiplicative property (b) in Theorem (3), we compute 1 det( ) = det ( ) B I P A I P = 1 det( P ) det( A I) det( P) ----(1) Slide
14 SIMILARITY det( P ) det( P) = det( P P) = det I = 1 det( B I) = det( A I) Since 1 1, we see from equation (1) that. Warnings: 1. The matrices and are not similar even though they have the same eigenvalues. Slide
15 SIMILARITY 2. Similarity is not the same as row equivalence. (If A is row equivalent to B, then B = EA for some invertible matrix E ). Row operations on a matrix usually change its eigenvalues. Slide
Eigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
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