Computation of eigenvalues and eigenvectors with respect to an inner product
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I have the square matrix $C=A^{-1}B$, where the square matrices $A, B$ are symmetric and positive definite. The matrix $A$ is also a Stieltjes tridiagonal matrix and the matrix $B$ is a triadiagonal matrix with positive elements in main, upper and down diagonal. We define a new inner product, that is
$$(v,w)_{B}:=Bvcdot w=sum_{i=1}^n(Bv)_iw_i,;;forall v,winmathbb{R}^n.$$
The matrix $C$ is symmetric, positive definite with respect to this inner product, so it has positive eigenvalues $leftlbracelambda_jrightrbrace_{i=1}^n$ and orthonormal eigenvectors $leftlbracephi_jrightrbrace_{i=1}^n$ with respect to this inner product. How can I compute this eigenvalues and eigenvectors by hand, for example with $n=4$ or for random $n$ with Matlab?
linear-algebra matrices eigenvalues-eigenvectors
edited Nov 20 at 10:28
Bernard
115k637107
asked Nov 20 at 10:15
math_lover
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I have the square matrix $C=A^{-1}B$, where the square matrices $A, B$ are symmetric and positive definite. The matrix $A$ is also a Stieltjes tridiagonal matrix and the matrix $B$ is a triadiagonal matrix with positive elements in main, upper and down diagonal. We define a new inner product, that is
$$(v,w)_{B}:=Bvcdot w=sum_{i=1}^n(Bv)_iw_i,;;forall v,winmathbb{R}^n.$$
The matrix $C$ is symmetric, positive definite with respect to this inner product, so it has positive eigenvalues $leftlbracelambda_jrightrbrace_{i=1}^n$ and orthonormal eigenvectors $leftlbracephi_jrightrbrace_{i=1}^n$ with respect to this inner product. How can I compute this eigenvalues and eigenvectors by hand, for example with $n=4$ or for random $n$ with Matlab?
linear-algebra matrices eigenvalues-eigenvectors
edited Nov 20 at 10:28
Bernard
115k637107
asked Nov 20 at 10:15
math_lover
9010
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have the square matrix $C=A^{-1}B$, where the square matrices $A, B$ are symmetric and positive definite. The matrix $A$ is also a Stieltjes tridiagonal matrix and the matrix $B$ is a triadiagonal matrix with positive elements in main, upper and down diagonal. We define a new inner product, that is
$$(v,w)_{B}:=Bvcdot w=sum_{i=1}^n(Bv)_iw_i,;;forall v,winmathbb{R}^n.$$
The matrix $C$ is symmetric, positive definite with respect to this inner product, so it has positive eigenvalues $leftlbracelambda_jrightrbrace_{i=1}^n$ and orthonormal eigenvectors $leftlbracephi_jrightrbrace_{i=1}^n$ with respect to this inner product. How can I compute this eigenvalues and eigenvectors by hand, for example with $n=4$ or for random $n$ with Matlab?
linear-algebra matrices eigenvalues-eigenvectors
edited Nov 20 at 10:28
Bernard
115k637107
asked Nov 20 at 10:15
math_lover
9010
I have the square matrix $C=A^{-1}B$, where the square matrices $A, B$ are symmetric and positive definite. The matrix $A$ is also a Stieltjes tridiagonal matrix and the matrix $B$ is a triadiagonal matrix with positive elements in main, upper and down diagonal. We define a new inner product, that is
$$(v,w)_{B}:=Bvcdot w=sum_{i=1}^n(Bv)_iw_i,;;forall v,winmathbb{R}^n.$$
The matrix $C$ is symmetric, positive definite with respect to this inner product, so it has positive eigenvalues $leftlbracelambda_jrightrbrace_{i=1}^n$ and orthonormal eigenvectors $leftlbracephi_jrightrbrace_{i=1}^n$ with respect to this inner product. How can I compute this eigenvalues and eigenvectors by hand, for example with $n=4$ or for random $n$ with Matlab?
linear-algebra matrices eigenvalues-eigenvectors
linear-algebra matrices eigenvalues-eigenvectors
edited Nov 20 at 10:28
Bernard
115k637107
asked Nov 20 at 10:15
math_lover
9010
edited Nov 20 at 10:28
Bernard
115k637107
asked Nov 20 at 10:15
math_lover
9010
edited Nov 20 at 10:28
Bernard
115k637107
edited Nov 20 at 10:28
Bernard
115k637107
edited Nov 20 at 10:28
Bernard
115k637107
115k637107
asked Nov 20 at 10:15
math_lover
9010
asked Nov 20 at 10:15
math_lover
9010
asked Nov 20 at 10:15
math_lover
9010
9010
add a comment |
add a comment |
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