【正文】 矩陣運算中的作用，以及部分在實際生活中的應(yīng)用。在例題解析中運用一些特征值與特征向量的性質(zhì)和方法，可以使問題更簡單，運算上更方便，量的例子加以說明運用特征值與特征向量的性質(zhì)可以使問題更加清楚，從而使高等代數(shù)中的大量習(xí)題迎刃而解，把特征值與特征向量在解決實際問題中的優(yōu)越性表現(xiàn)出來.28河北師范大學(xué)匯華學(xué)院本科生畢業(yè)論文（設(shè)計）翻譯文章矩陣的特征值可以確定所發(fā)現(xiàn)的特征多項式的根。多項式的根的顯式代數(shù)公式僅當(dāng)存在比率為4以下。根據(jù)阿貝爾 魯菲尼定理5個或5個以上的多項式的根源是沒有一般情況下，明確和準(zhǔn)確的代數(shù)公式。事實證明，任何程度的多項式是一些同伴階矩陣的特征多項式。因此，5個或更多的順序的矩陣的特征值和特征向量不能獲得通過明確的代數(shù)公式，因此，必須計算的近似數(shù)值方法在理論上，可以精確計算的特征多項式的系數(shù)，因為它們是矩陣元素的總和，有算法，可以找到任何所需的精度。然而，任意程度的多項式的所有根這種方法在實踐中是不可行的，因為系數(shù)將被污染的不可避免的舍入誤差，多項式的根可以是一個極為敏感的功能（例如由威爾金森的多項式系數(shù)） 。在實踐中可行，因為系數(shù)將被污染的不可避免的舍入誤差，多項式的根可以是一個極為敏感的功能（例如由威爾金森的多項式系數(shù)）直到 QR 算法在1961年的來臨，高效，精確的方法來計算任意矩陣的特征值和特征向量。 [與 LU 分解法的查詢結(jié)果在一個算法中與更好地的 QR 算法的收斂性比。結(jié)合了 Householder 變換。對于大的的厄密共軛的稀疏矩陣，theLanczos 算法。 是一個有效的迭代的方法，以計算特征值和特征向量獲得的一個例子，在一些其他的可能性。[編輯]計算特征向量一旦一個特征值（精確）的值是已知的，可以找到對應(yīng)的特征向量，通過尋找特征值方程的非零解，即成為與已知的系數(shù)的線性方程系統(tǒng)。例如，一旦它是已知的，圖6是矩陣的特征值我們可以找到它的特征向量，通過求解方程，也就是 ?????????????yx6314該矩陣方程相當(dāng)于兩個線性方程組的 也就是?????6y3x4?????02x兩個方程減少到單一的線性方程 .因此，任何載體的形式，任何非零實數(shù)，是一個y2特征值與特征向量相匹配。上述矩陣 A 有另一個特征值。類似的計算表明，對應(yīng)的特征向量是非零的解決方案，那就是，任何載體的形式，任何非零實數(shù) b。某些數(shù)字的方法，計算的矩陣的特征值也確定一組對應(yīng)的特征向量作為副產(chǎn)物的計算。里昂，線性代數(shù)（第一版） 【M】.北京：機械工業(yè)出版社， 2022.The eigenvalues of a matrix can be determined by finding the roots of the characteristic polynomial. Explicit algebraic formulas for the roots of a polynomial exist only if the degree is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more.It turns out that any polynomial with degree is the characteristic polynomial of some panion matrix of order . Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be 29puted by approximate numerical methods.In theory, the coefficients of the characteristic polynomial can be puted exactly, since they are sums of products of matrix elements。 and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy.[10] However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable roundoff errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson39。s polynomial).[10]Efficient, accurate methods to pute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the QR algorithm in 1961. [10] Combining the Householder transformation with the LU deposition results in an algorithm with better convergence than the QR algorithm.[11] For large Hermitian sparse matrices, theLanczos algorithm is one example of an efficient iterative method to pute eigenvalues and eigenvectors, among several other possibilities.[10][edit] Computing the eigenvectorsOnce the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that bees a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix ???????3614Awe can find its eigenvectors by solving the equation ， that isV6A??????????????yx6314This matrix equation is equivalent to two linear equations that is ????yx4?????02xBoth equations reduce to the single linear equation . Therefore, any vector of the form y2?,for any nonzero real number a, is an eigenvector of A with eigenvalue .??39。2,a 6??The matrix A above has another eigenvalue . A similar calculation shows that the 1?corresponding eigenvectors are the nonzero solutions of , that is, any vector of the 03??yxform , for any nonzero real number .??39。3b?Some numeric methods that pute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a byproduct of the putation.