Symmetric matrices and positive definiteness

Symmetric matrices are good - their eigenvalues are real and each has a complete set of orthonormal eigenvectors. Positive definite matrices are even better.

对称矩阵的性质很不错–它们的特征向量是实值的 完备正交特征向量集合
正定矩阵，具有更加优秀的性质

Symmetric matrices

A symmetric matrix is one for which.
If a matrix has some special property (e.g. it’s a Markov matrix), its eigenvalues and eigenvectors are likely to have special properties as well.

如果矩阵有一些特殊的性质（比如说马尔可夫矩阵），它的特征值和特征向量更可能拥有特别性质

For a symmetric matrix with real number entries, the eigenvalues are real numbers and it’s possible to choose a complete set of eigenvectors that are perpendicular (or even orthonormal).

对于实值对称矩阵，特征值是实值，且有可能性选择一组相互正交的特征向量。

Ifhasindependent eigenvector we can write. Ifis symmetric we can write, whereis an orthogonal matrix. Mathematicians call this the spectral theorem and think of the eigenvalues as the “spectrum” and think of the eigenvalues as the “spectrum” of the matrix. In mechanics it’s called the principal axis theorem.

In addition, any matrix of the formwill be symmetric.

Real eigenvalues

Why are the eigenvalues of a symmetric matrix real? Supposeis symmetric and. Then we can conjugate to get. If the entries ofare real, this becomes. (This proves that complex eigenvalues of real valued matrices come in conjugate pairs.)

Now transpose to get. Becauseis symmetric we now have. Multiplying both sides of this equation on the right bygives:

On the other hand, we can multiplyon the left byto get:

Comparing the two equations we see thatand, unlessis zero, we can concludeis real.

How do we know?

Ifthen.

With complex vectors, as with complex numbers, multiplying by the conjugate is often helpful.
Symmetric matrices with real entries, then it will have real eigenvalues and perpendicular eigenvectors if an only if. (The proof of this follows the same pattern.)

Projection onto eigenvectors

If, we can write:

The matrixis the projection matrix onto, so every symmetric matrix is a combination of perpendicular projection matrices.

Information about eigenvalues

If we know that eigenvalues are real, we can ask whether they are positive or negative. (Remember that the signs of the eigenvalues are important in solving systems of differential equations.)
For very large matices, it’s impractical to compute eigenvalues by solving. However, it’s not hard to compute the pivots, and the signs of the pivots of a symmetric matrix are the same as the signs of the eigenvalues:

Because the eigenvalues ofare justmore than the eigenvalues of, we can use this fact to find which eigenvalues of a symmetric matrix are greater or less than any real number. This tells us a lot about the eigenvalues ofeven if we can;t compute them directly.

Positive definite matrices

A positive definite matrix is a symmetric matrixfor which all eigenvalues are positive. A good way to tell if a matrix is positive definite is to check that all its pivots are positive.

Let. The pivots of this matrix are 5 and. The matrix is symmetric and its pivots (and therefore eigenvalues) are positive, sois a positive definite matrix. Its eigenvalues are the solutions to:

The determinant of a positive definite matrix is always positive but the determinant ofis also positive, and that matrix isn’t positive definite. If all of the subdeterminants ofare positive (determinants of thebymatrices in the upper left corner of, where), thenis positive definite.

The subject of positive definite matrices brings together what we’ve learned about pivots, determinants and eigenvalues of square matrices. Soon we’ll have a chance to bring together what we’ve learned in this course and apply it to non-square matrices.