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HomeMatlabWhat Is an Eigenvalue? – Nick Higham

What Is an Eigenvalue? – Nick Higham

An eigenvalue of a sq. matrix $A$ is a scalar $lambda$ such that $Ax = lambda x$ for some nonzero vector $x$. The vector $x$ is an eigenvector of $A$ and it has the excellence of being a route that isn’t modified on multiplication by $A$.

An $ntimes n$ matrix has $n$ eigenvalues. This may be seen by noting that $Ax = lambda x$ is equal to $(lambda I - A) x = 0$, which signifies that $lambda I - A$ is singular, since $xne 0$. Therefore $det(lambda I - A) = 0$. However

$notag det(lambda I - A) = lambda^n + a_{n-1}lambda^{n-1} + cdots + a_1 lambda + a_0$

is a scalar polynomial of diploma $n$ (the attribute polynomial of $A$) with nonzero main coefficient and so has $n$ roots, that are the eigenvalues of $A$. Since $det(lambda I - A) = det( (lambda I - A)^T) = det(lambda I - A^T)$, the eigenvalues of $A^T$ are the identical as these of $A$.

An actual matrix might have advanced eigenvalues, however they seem in advanced conjugate pairs. Certainly $Ax = lambda x$ implies $overline{A}overline{x} = overline{lambda} overline{x}$, so if $A$ is actual then $overline{lambda}$ is an eigenvalue of $A$ with eigenvector $overline{x}$.

Listed below are some $2times 2$ matrices and their eigenvalues.

notag begin{aligned} A_1 &= begin{bmatrix}1 & 0 0 & 1 end{bmatrix}, quad lambda = 1,1, A_2 &= begin{bmatrix}0 & 1 0 & 0 end{bmatrix}, quad lambda = 0,0, A_3 &= begin{bmatrix}0 & 1 -1 & 0 end{bmatrix}, quad lambda = mathrm{i},mathrm{-i}. end{aligned}

Be aware that $A_1$ and $A_2$ are higher triangular, that’s, $a_{ij} = 0$ for $i>j$. For such a matrix the eigenvalues are the diagonal components.

A symmetric matrix ($A^T = A$) or Hermitian matrix ($A^* = A$, the place $A^* = overline{A}^T$) has actual eigenvalues. A proof is $Ax = lambda x Rightarrow x^*A^* = overline{lambda} x^*$ so premultiplying the primary equation by $x^*$ and postmultiplying the second by $x$ provides $x^*Ax = lambda x^*x$ and $x^*Ax = overline{lambda} x^*x$, which signifies that $(lambda-overline{lambda})x^*x = 0$, or $lambda=overline{lambda}$ since $x^*x ne 0$. The matrix $A_1$ above is symmetric.

A skew-symmetric matrix ($A^T = -A$) or skew-Hermitian advanced matrix ($A^* = -A$) has pure imaginary eigenvalues. A proof is just like the Hermitian case: $Ax = lambda x Rightarrow -x^*A = x^*A^* = overline{lambda} x^*$ and so $x^*Ax$ is the same as each $lambda x^*x$ and $-overline{lambda} x^*x$, so $lambda = -overline{lambda}$. The matrix $A_3$ above is skew-symmetric.

Normally, the eigenvalues of a matrix $A$ can lie anyplace within the advanced airplane, topic to restrictions primarily based on matrix construction equivalent to symmetry or skew-symmetry, however they’re restricted to the disc centered on the origin with radius $|A|$, as a result of for any matrix norm $|cdot|$ it may be proven that each eigenvalue satisfies $|lambda| le |A|$.

Listed below are some instance eigenvalue distributions, computed in MATLAB. (The eigenvalues are computed at excessive precision utilizing the Advanpix Multiprecision Computing Toolbox in an effort to be sure that rounding errors don’t have an effect on the plots.) The second and third matrices are actual, so the eigenvalues are symmetrically distributed about the true axis. (The primary matrix is advanced.)

Though this text is about eigenvalues we have to say somewhat extra about eigenvectors. An $ntimes n$ matrix $A$ with distinct eigenvalues has $n$ linearly impartial eigenvectors. Certainly it’s diagonalizable: $A = XDX^{-1}$ for some nonsingular matrix $X$ with $D = mathrm{diag}(lambda_i)$ the matrix of eigenvalues. If we write $X$ when it comes to its columns as $X = [x_1,x_2,dots,x_n]$ then $AX = XD$ is equal to $Ax_i = lambda _i x_i$, $i=1colon n$, so the $x_i$ are eigenvectors of $A$. The matrices $A_1$ and $A_3$ above each have two linearly impartial eigenvectors.

If there are repeated eigenvalues there will be lower than $n$ linearly impartial eigenvectors. The matrix $A_2$ above has just one eigenvector: the vector $left[begin{smallmatrix}1 0 end{smallmatrix}right]$ (or any nonzero scalar a number of of it). This matrix is a Jordan block. The matrix $A_1$ exhibits {that a} matrix with repeated eigenvalues can have linearly impartial eigenvectors.

Listed below are some questions on eigenvalues.

• What matrix decompositions reveal eigenvalues? The reply is the Jordan canonical type and the Schur decomposition. The Jordan canonical type exhibits what number of linearly impartial eigenvectors are related to every eigenvalue.
• Can we get hold of higher bounds on the place eigenvalues lie within the advanced airplane? Many outcomes can be found, of which probably the most well-known is Gershgorin’s theorem.
• How can we compute eigenvalues? Varied strategies can be found. The QR algorithm is extensively used and is relevant to all sorts of eigenvalue issues.

Lastly, we notice that the idea of eigenvalue is extra basic than simply for matrices: it extends to nonlinear operators on finite or infinite dimensional areas.

References

Many books embrace therapies of eigenvalues of matrices. We give simply three examples.

• Gene Golub and Charles F. Van Mortgage, Matrix Computations, fourth version, Johns Hopkins College Press, Baltimore, MD, USA, 2013.
• Roger A. Horn and Charles R. Johnson, Matrix Evaluation, second version, Cambridge College Press, 2013. My assessment of the second version.
• Carl D. Meyer, Matrix Evaluation and Utilized Linear Algebra, Society for Industrial and Utilized Arithmetic, Philadelphia, PA, USA, 2000.
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