Symmetric Matrix Diagonalizable Proof

16 Apr, 2021

Symmetric and hermitian matrices which arise in many applications enjoy the property of always being diagonalizable. Then A is orthogonal ß A1 AT ß In ATA MATH 316U 003 - 83 Diagonalization of Symmetric Matrices1.

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Every Hermitian matrix is diagonalizable.

Symmetric matrix diagonalizable proof. A real matrix Ais symmetric if and only if Acan be diagonalized by an orthogonal matrix ie. Real symmetric matrices are diagonalizable by orthogonal matrices. 83 Diagonalization of Symmetric Matrices DEFp368 A is called an orthogonal matrix if A1 AT.

22 Diagonalizability of symmetric matrices The main theorem of this section is that every real symmetric matrix is not only diagonalizable but orthogonally diagonalizable. This is sometimes written as u v. Equivalently a square matrix is symmetric if and only if there exists anorthogonal matrixSsuchthatSTASis diagonal.

Let 1 and 2 be distinct eigenvalues of A with Av 1 1 v 1. Real-valued symmetric matrices always diagonalize Vera Sacrist an De nition 1 The sum of two subspaces W 1 and W 2 of a vector space V is de ned as W W 1 W 2 fw2V jw w 1 w 2w 1 2W 1w 2 2W 2g. Sketch of proof of converse.

Proposition An orthonormal matrix P has the property that P1 PT. More generally matrices are diagonalizable by unitary matrices if and only if they are normal. A matrix P is said to be orthonormal if its columns are unit vectors and P is orthogonal.

In particular they areorthogonallydiagonalizable. In other words U is orthogonal if U-1 UmathsfT. Assume symmetric R M commute.

Symmetric matrix A meaning A AT. U 1AUH UHAHU 1H U 1AU. Diagonalization of Symmetric Matrices Let A 2Rn n be a symmtric matrix.

Theorem If A is a real symmetric matrix then there exists an orthonormal matrix P such that i P1AP D where D a diagonal matrix. In particular every real symmetric matrix is diagonalizable. That is a matrix is orthogonally diagonalizable if and only if it is symmetric.

This implies 2 1 vT 2 v 1 0 or v T 2 v 1 0. So that 1 v T 2 v 1 v T 2 Av 1 Av 2 v 1 2 v T 2 v 1. A matrix A in MnR is called orthogonal if.

So such a diagonalization is not necessarily unique. 53 as we will see in our examples. Symmetric matrices have very nice properties.

We say that U in mathbbRntimes n is orthogonal if UmathsfTU UUmathsfT I_n. 2 Ais orthogonally diagonalizable. The identity matrix is symmetric and is diagonalizable by any invertible matrix P because P 1 I P I.

Lemma 1 The sum W 1 W 2 of two subspaces of a vector space V is a subspace of V. A UDU 1 with Uorthogonal and Ddiagonal. Theorem 1 The spectral theorem.

Two vectors u and v in Rn are orthogonal to each other if uv 0 or equivalently if uTv 0. So A should be symmetric. Also the set of eigenvectors of such matrices can always be chosen as orthonormal.

Therefore the theorem is called theSpectral Theorem for real symmetric matrices. We are now ready to prove our main theorem. Clearly if A is 1times 1 we can simply set U 1 and D A.

Assume R UDUT so diagonal entries of D are eigenvalues of R and columns of U are eigenvectors of R. Diagonalization of symmetric matrices Theorem. For the induction hypothesis assume that every n-1times n.

Then RM MR UDUTM MUDUT DUTMU UTMUD since U orthogonal. Av 2 2 v 2. This means that if Ais symmetric there is a basis Bv1 vnforRnconsisting of eigenvectors forAso that the vectors inBare pairwise orthogonalAnother way of saying this is that there exists a matrixPwith real entries such that.

By the above theorem Ais triangularizablethat is we can nd a unitary matrix Usuch that U 1AU T with Tupper triangular. A 2 4 1 1 1 1 1 1 1 1 1 3 5. Then the eigenvectors of T are all real and there is an orthonormal basis V v1vn of Rn consisting of eigenvectors of T so that MtxWT is diagonalizable.

To illustrate the theorem let us diagonalize the following matrix by an orthogonal matrix. Gexin Yu gyuwmedu Section 71 Diagonalization of symmetric. The set of eigenvalues of a matrix is sometimescalled the spectrum of the matrix and orthogonal diagonalization of a matrix E factors Ein away that displays all the eigenvalues and their multiplicities.

MtxWT MtxWV idMtxV TMtxVWid where MtxV T is diagonal and the change of basis matrices MtxVWid and MtxWV id. If A is symmetric then it has an orthonormal basis d 1 d 2 d n of column eigenvectors with corresponding eigenvalues λ. Real symmetric matrices not only have real eigenvalues they are always diagonalizable.

Square matrix is symmetric if and only if it has an orthonormal eigenbasis. The diagonalization procedure is essentially the same as outlined in Sec. The converse also holds so symmetric matrices commute if and only if they are simultaneously diagonalizable.

For each item nd an explicit example or explain why none exists. Any two real eigenvectors pertaining to two distinct real eigenvalues of A are orthogonal. TheoremAn n n matrix A is orthogonally diagonalizableif and only if A is symmetric.

If W 1 W 2 f0g the sum is called direct and we write W 1 W 2. Here is a shortcut to nd the eigenvalues. Let A be an ntimes n real symmetric matrix.

Let Abe a Hermitian matrix. It is a beautiful story which carries the beautiful name the spectral theorem. TH 88p369 A is orthogonal if and only if the column vectors of A form an orthonormal set.

In fact more can be said about the diagonalization. We prove that A is orthogonally diagonalizable by induction on the size of A. ProofLetA u1 un.

Ie given a real symmetric matrix is diagonal for some orthogonal matrix. If Ais an n nsym-metric matrix then 1All eigenvalues of Aare real.

Linear Algebra Example Problems Diagonalizing A Matrix Youtube