Linear Algebra and Matrices


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1 LECTURE Linear Algebra and Matrices Before embarking on a study of systems of differential equations we will first review, very quickly, some fundamental objects and operations in linear algebra.. Matrices Definition.. An n m matrix n by m matrix ) is an arrangement of nm objects usually numbers) into a rectangular array with n rows and m columns. We ll typically denote the entry in the i th row and j th column of a matrix A as a ij and similarly, b ij for the ij) th entry of a matrix B). Thus, for example, a a a m a a a m A..... a n a n a nm An n matrix is called a column vector and a m matrix is called a row vector. v v v. v n v v, v,..., v m.. Matrix Operations. Definition.. The sum of two n m matrices A and B is the n m matrix A + B with entries A + B) ij a ij + b ij. Definition.3. The scalar product of an n m matrix A with a number λ is the n m matrix λa with entries λa) ij λa ij. Definition.4. The difference of two n m matrices A and B is the n m matrix A B with entries A B) ij a ij b ij A + ) B) ij Definition.5. The transpose of an n m matrix A is the m n matrix A t with entries A t ) ij A ji Note the i th row of A t is simply the j th column of A written horizontally.) Definition.6. An n n matrix A is said to be symmetric if A A t.
2 . COMPLEX NUMBERS Note that the transpose of a rowvector is a column vector and the transpose of a column vector is a row vector. Definition.7. The dot product of an ndimensional row vector r,..., r n with a ndimensional column c vector. is the number c n r,..., r n c.. r c + + r n c n c n This coincides with the usual dot products of vectors when we think of vectors as ordered lists of numbers rather than special kinds of matrices.) Definition.8. The matrix product of an n m matrix A with a m q matrix B is the n q matrix with entries m AB) ij a ik b kj Note that the ij) th entry of the matrix product AB is the dot product of the i th row of A with the j th column of B. k. Complex Numbers In what follows it will be sometime necessary to consider vectors and matrices with entries that are complex numbers. We recall here some basic facts about complex numbers. Definition.9. A complex numbers are pairs x, y) of real numbers which satisfying the following addition and multiplication rules x, y) + x, y ) x + x, y + y ) x, y) x, y ) xx yy, xy + yx ) Usually though we write a complex number as z x + iy and then add and multiply complex numbers by employing the usual rules of arithmetic and the relation i. Thus, x + iy) x + iy ) xx + x iy ) + iy x ) + iy) iy ) xx yy + i xy + yx ) When we write z x + iy we say that x is the real part of z and y is the imaginary part of z. The complex conjugate of z x + iy is the complex number z obtained by switching the sign of it imaginary part z x iy We have Re z) x z + z Im z) y z z i Re )
3 3. DETERMINANTS 3 We will also sometimes consider functions of a complex variable. Here we only state the very fundamental Euler Formula We note that e x+iy e x cos y) + i sin y)) cos y) Re e iy) sin y) Im e iy) Definition.. A complex matrix is a matrix whose entries are complex numbers. Similarly, a complex vector is a vector whose components are complex numbers. The hermitian adjoint of a complex matrix A is the matrix A with entries A ) ij a ji Equivalently, A A ) t At ) Example.. A + i i 3 + i 3 ) A i 3 i + i 3 ) Definition.. An n n matrix is hermitian or selfadjoint) if A A Definition.3. Let x and y be two complex column) vectors. The hermitian) inner product x, y) of x and y is the complex number x t y x,, x n y. y n x y + + x n y n Remark.4. If A is a real hermitian matrix that is, all of its entries are real numbers), then A A ) t A) t A t so a real hemitian matrix is just a real symmetric matrix. If x and y are two real vectors then the hermitian inner product x, y) coincides with the usual dot product of real vectors. 3. Determinants Definition.5. The ij) th minor of an n n matrix A is the n ) n ) matrix M ij obtained by deleting the i th row and j th column from A. Definition.6. The determinant of an n n matrix A is the number det A determined by the following recursive algorithm: If A is a matrix a, then det A a if A is an n n matrix then n det A : ) i+j a ij det M ij : j n ) i+j a ij det M ij i i any fixed row index) j any fixed column index)
4 5. EIGENVALUES AND EIGENVECTORS 4 The following formulas for the determinants of and 3 3 matrices follow easily from the above definition a a det ) + a a a det M ) + ) + a det M ) det a a a 3 a a a 3 a 3 a 3 a 33 a det a a det a a a a a ) + a det a a 3 a 3 a 33 + ) +3 a a a 3 det a 3 a 3 + ) + a a a det 3 a 3 a 33 a a a 33 a 3 a 3 ) a a a 33 a 3 a 3 ) + a 3 a a 3 a a 3 ) 4. Inverses Definition.7. The inverse of an n n matrix A is the n n matrix A such that N.B. matrix inverses do not always exist.) A A I AA There are two basic ways of computing matrix inverses: Row reduction: A I row reduction I A Cofactor Method: A det A Ct where C is the cofactor matrix of A whose entries are ± times the determinants of the n ) n ) minors of A. C) ij ) i+j det M ij ) 5. Eigenvalues and Eigenvectors Definition.8. Let A be an n n matrix. If there exists a number λ and an ndimensional column) vector v such that Av λv then v is said to be an eigenvector of A and λ is said to be the eigenvalue of A corresponding to the eigenvector v. The following algorithm determines all the eigenvectors and eigenvalues of an n n matrix A. Set p A det A λi). Here λ is regarded as a variable, and λi is the matrix obtained by scalar multiplying the n n identity matrix by λ. A λi is thus the matrix obtained by subtracting λ from each of the diagonal entries of A. Upon computation, det A λi) will yield a polynomial of degree n in λ. The solutions of will be the eigenvalues of A. p A λ)
5 5. EIGENVALUES AND EIGENVECTORS 5 For each root λ r of p A λ), find the general solution of A ri) x and expand that solution in terms of the free parameters of the solution. The eigenvectors corresponding to the eigenvalue r of A will be the constant vectors in that expansion. Example.9. Find the eigenvalues and eigenvectors of A. We have p A λ) : det λ λ λ) 4 λ λ 3 λ 3) λ + ) Since λ 3, are the solutions of p A λ), these are the eigenvalues of A. Now we look for the eigenvectors corresponding to λ 3. 3 x x 3 The reduced row echelon form of the coefficient matrix is { x x x x x so v λ3 x Similarly, we look for the eigenvectors corresponding to λ : ) x x ) x x and so Example.. Suppose Find the eigenvectors and eigenvalues of A. v λ A ) x The characteristic polynomial is λ p A λ) det λ) + λ λ + λ To solve p A λ), we apply the quadratic formula and find a + bx + c x b ± b 4ac a p A λ) λ ± 4 8 Thus we have two complex eigenvalues. ± 4 ± 4 ± i ± i
6 6. DIAGONALIZATION OF MATRICES 6 Setting λ + i, we now solve + i) x + i) i x i x The row reduction method of solving linear systems still works: i i R i R + ir R i) R ix i x i x v λ+i Similarly, for λ i, one finds i v λ i i i 6. Diagonalization of Matrices Recall that a diagonal matrix is a square n n matrix with nonzero entries only along the diagonal from the under left to the lower right the main diagonal). Diagonal matrices are particularly convenient for eigenvalue problems since the eigenvalues of a diagonal matrix a A a..... a nn coincide with the diagonal entries {a ii } and the eigenvector corresponding the eigenvalue a ii is just the i th coordinate vector. That is, if A is of the above form, we always have Ae i a ii e i Definition.. An n n matrix A is diagonalizable if there is an invertible n n matrix C such that C AC is a diagonal matrix. The matrix C is said to diagonalize A. Lemma.. Let A be a real or complex) n n matrix, let λ, λ,..., λ n be a set of n real respectively, complex) scalars, and let v, v,..., v n be a set of n vectors in R n respectively, C n ). Let C be the n n matrix formed by using v j for j th column vector, and let D be the n n diagonal matrix whose diagonal entries are λ, λ,..., λ n. Then AC CD if and only if λ, λ,..., λ n are the eigenvalues of A and each v j is an eigenvector of A correponding the eigenvalue λ j. Now suppose AC CD, and the matrix C is invertible. D C AC. Then we can write And so we can think of the matrix C as converting A into a diagonal matrix. Theorem.3. An n n matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. Example.4. Find the matrix that diagonalizes 6 A
7 6. DIAGONALIZATION OF MATRICES 7 First we ll find the eigenvalues and eigenvectors of A. λ 6 det A λi) det λ) λ) λ, λ The eigenvectors corresponding to the eigenvalue λ are solutions of A )I) x or 6 x 6x x 3 3 x r The eigenvectors corresponding to the eigenvalue λ are solutions of A )I) x or 3 6 x 3x + 6 x x r So the vectors v, and v, will be eigenvectors of A. We now arrange these two vectors as the column vectors of the matrix C. C In order to compute the diagonalization of A we also need C. This we compute using the technique of Section.5: R R + R C Finally, D C AC C AC) ) 6
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