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In general, I am unsure that the spectral norm is the most widely used. For example the Frobenius norm is used for to approximate solution on non-negative matrix factorisation or correlation/covariance matrix regularisation. I think that part of this question stems from the terminology misdemeanour some people do (myself included) when referring to the Frobenius norm as the Euclidean matrix norm. We should not because actually the $L_2$ matrix norm (ie. the spectral norm) is the one that is induced to matrices when using the $L_2$ vector norm. The Frobenius norm is that is element-wise: $||A||_F = \sqrt{\sum_{i,j}a_{i,j}^2}$, while the $L_2$ matrix norm ($||A||_2 = \sqrt{\lambda_{max}(A^T A)})$) is based on singular values so it is therefore more "univeral". (for luck of a better term?) The $L_2$ matrix norm is a Euclidean-type norm since it is induced by the Euclidean vector norm, where $||A||_2 = \max\limits_{||x||_2 =1} || Ax||_2$. It therefore an induced norm for matrices because it is induced by a vector norm, the $L_2$ vector norm in this case.

Probably MATLAB aims to provide the $L_2$ norm by default when using the command norm; as a consequence it provides the Euclidean vector norm but also the $L_2$ matrix norm, ie. the spectral matrix norm (rather than the wrongly quoted "Frobenius/Euclidean matrix norm"). Finally let me note that what is the default norm is a matter of opinion to some extend: For example J.E. Gentle's "Matrix Algebra - Theory, Computations, and Applications in Statistics" literally has a chapter (3.9.2) named: "The Frobenius Norm - The “Usual” Norm"; so clearly the spectral norm is not the default norm for all parties considered! :) As commented by @amoeba, different communities might have different terminology conventions. It goes without saying that I think Gentle's book is an invaluable resource on the matter of Lin. Algebra application in Statistics and I would prompt you to look it further!

In general, I am unsure that the spectral norm is the most widely used. For example the Frobenius norm is used for to approximate solution on non-negative matrix factorisation or correlation/covariance matrix regularisation. I think that part of this question stems from the terminology misdemeanour some people do (myself included) when referring to the Frobenius norm as the Euclidean matrix norm. We should not because actually the $L_2$ matrix norm (ie. the spectral norm) is the one that is induced to matrices when using the $L_2$ vector norm. The Frobenius norm is that is element-wise: $||A||_F = \sqrt{\sum_{i,j}a_{i,j}^2}$, while the $L_2$ matrix norm ($||A||_2 = \sqrt{\lambda_{max}(A^T A)})$) is based on singular values so it is therefore more "universal" (for lack of a better term?). The $L_2$ matrix norm is a Euclidean-type norm since it is induced by the Euclidean vector norm, where $||A||_2 = \max\limits_{||x||_2 =1} || Ax||_2$. It therefore an induced norm for matrices because it is induced by a vector norm, the $L_2$ vector norm in this case.

Probably MATLAB aims to provide the $L_2$ norm by default when using the command norm; as a consequence it provides the Euclidean vector norm but also the $L_2$ matrix norm, ie. the spectral matrix norm (rather than the wrongly quoted "Frobenius/Euclidean matrix norm"). Finally let me note that what is the default norm is a matter of opinion to some extend: For example J.E. Gentle's "Matrix Algebra - Theory, Computations, and Applications in Statistics" literally has a chapter (3.9.2) named: "The Frobenius Norm - The “Usual” Norm"; so clearly the spectral norm is not the default norm for all parties considered! :) As commented by @amoeba, different communities might have different terminology conventions. It goes without saying that I think Gentle's book is an invaluable resource on the matter of Linear Algebra application in Statistics and I would prompt you to look it further!

In general, I am unsure that the spectral norm is the most widely used. For example the Frobenius norm is used for to approximate solution on non-negative matrix factorisation or correlation/covariance matrix regularisation. I think that part of this question stems from the terminology misdemeanour some people do (myself included) when referring to the Frobenius norm as the Euclidean matrix norm. We should not because actually the $L_2$ matrix norm (ie. the spectral norm) is the one that is induced to matrices when using the $L_2$ vector norm. The Frobenius norm is that is element-wise: $||A||_F = \sqrt{\sum_{i,j}a_{i,j}^2}$, while the $L_2$ matrix norm ($||A||_2 = \sqrt{\lambda_{max}(A^T A)})$) is based on singular values so it is therefore more "univeral". (for luck of a better term?) The $L_2$ matrix norm is a Euclidean-type norm since it is induced by the Euclidean vector norm, where $||A||_2 = \max\limits_{||x||_2 =1} || Ax||_2$. It therefore an induced norm for matrices because it is induced by a vector norm, the $L_2$ vector norm in this case.

Probably both MATLAB and R aim to provide the $L_2$ norm by default when using the command norm; as a consequence they provide the Euclidean vector norm but also the $L_2$ matrix norm, ie. the spectral matrix norm (rather than the wrongly quoted "Frobenius/Euclidean matrix norm"). Finally let me note that what is the default norm is a matter of opinion to some extend: For example J.E. Gentle's "Matrix Algebra - Theory, Computations, and Applications in Statistics" literally has a chapter (3.9.2) named: "The Frobenius Norm - The “Usual” Norm"; so clearly the spectral norm is not the default norm for all parties considered! :) As commented by @amoeba, different communities might have different terminology conventions. It goes without saying that I think Gentle's book is an invaluable resource on the matter of Lin. Algebra application in Statistics and I would prompt you to look it further!

In general, I am unsure that the spectral norm is the most widely used. For example the Frobenius norm is used for to approximate solution on non-negative matrix factorisation or correlation/covariance matrix regularisation. I think that part of this question stems from the terminology misdemeanour some people do (myself included) when referring to the Frobenius norm as the Euclidean matrix norm. We should not because actually the $L_2$ matrix norm (ie. the spectral norm) is the one that is induced to matrices when using the $L_2$ vector norm. The Frobenius norm is that is element-wise: $||A||_F = \sqrt{\sum_{i,j}a_{i,j}^2}$, while the $L_2$ matrix norm ($||A||_2 = \sqrt{\lambda_{max}(A^T A)})$) is based on singular values so it is therefore more "univeral". (for luck of a better term?) The $L_2$ matrix norm is a Euclidean-type norm since it is induced by the Euclidean vector norm, where $||A||_2 = \max\limits_{||x||_2 =1} || Ax||_2$. It therefore an induced norm for matrices because it is induced by a vector norm, the $L_2$ vector norm in this case.

Probably MATLAB aims to provide the $L_2$ norm by default when using the command norm; as a consequence it provides the Euclidean vector norm but also the $L_2$ matrix norm, ie. the spectral matrix norm (rather than the wrongly quoted "Frobenius/Euclidean matrix norm"). Finally let me note that what is the default norm is a matter of opinion to some extend: For example J.E. Gentle's "Matrix Algebra - Theory, Computations, and Applications in Statistics" literally has a chapter (3.9.2) named: "The Frobenius Norm - The “Usual” Norm"; so clearly the spectral norm is not the default norm for all parties considered! :) As commented by @amoeba, different communities might have different terminology conventions. It goes without saying that I think Gentle's book is an invaluable resource on the matter of Lin. Algebra application in Statistics and I would prompt you to look it further!

For starters I think that part of this question stems from the terminology misdemeanour some people do (myself included) when referring to the Frobenius norm as the Euclidean matrix norm. We should not because actually the $L_2$ matrix norm (ie. the spectral norm) is the one that is induced to matrices when using the $L_2$ vector norm.

In general, I am unsure that the spectral norm is the most widely used. For example the Frobenius norm is used for to approximate solution on non-negative matrix factorisation or correlation/covariance matrix regularisation. A generic aspect of the Frobenius norm is that is element-wise ($||A||_F = \sqrt{\sum_{i,j}a_{i,j}^2}$) while the $L_2$ matrix norm ($||A||_2 = \sqrt{\lambda_{max}(A^T A)})$) is somewhat "more holistic". (for luck of a better term?) The $L_2$ matrix norm is a Euclidean-type norm since it is induced by the Euclidean vector norm, where $||A||_2 = \max\limits_{||x||_2 =1} || Ax||_2$. It therefore an induced norm for matrices because it is induced by a vector norm.

Let me note that probably both MATLAB and R aim to provide the $L_2$ norm by default when using the command norm; as a consequence they provide the Euclidean vector norm but also the $L_2$ matrix norm, ie. the spectral matrix norm (rather than the wrongly quoted "Frobenius/Euclidean matrix norm"). Finally let's note that what is the default norm is a matter of opinion to some extend: For example Gentle's "Matrix Algebra - Theory, Computations, and Applications in Statistics" literally has a chapter (3.9.2) named: "The Frobenius Norm - The “Usual” Norm"; so clearly the spectral norm is not the default norm for all parties considered! :) As commented by @amoeba, different communities might have different terminology conventions. It goes without saying that I think Gentle's book is an invaluable resource on the matter and I would prompt you to look it further!

In general, I am unsure that the spectral norm is the most widely used. For example the Frobenius norm is used for to approximate solution on non-negative matrix factorisation or correlation/covariance matrix regularisation. I think that part of this question stems from the terminology misdemeanour some people do (myself included) when referring to the Frobenius norm as the Euclidean matrix norm. We should not because actually the $L_2$ matrix norm (ie. the spectral norm) is the one that is induced to matrices when using the $L_2$ vector norm. The Frobenius norm is that is element-wise: $||A||_F = \sqrt{\sum_{i,j}a_{i,j}^2}$, while the $L_2$ matrix norm ($||A||_2 = \sqrt{\lambda_{max}(A^T A)})$) is based on singular values so it is therefore more "univeral". (for luck of a better term?) The $L_2$ matrix norm is a Euclidean-type norm since it is induced by the Euclidean vector norm, where $||A||_2 = \max\limits_{||x||_2 =1} || Ax||_2$. It therefore an induced norm for matrices because it is induced by a vector norm, the $L_2$ vector norm in this case.

Probably both MATLAB and R aim to provide the $L_2$ norm by default when using the command norm; as a consequence they provide the Euclidean vector norm but also the $L_2$ matrix norm, ie. the spectral matrix norm (rather than the wrongly quoted "Frobenius/Euclidean matrix norm"). Finally let me note that what is the default norm is a matter of opinion to some extend: For example J.E. Gentle's "Matrix Algebra - Theory, Computations, and Applications in Statistics" literally has a chapter (3.9.2) named: "The Frobenius Norm - The “Usual” Norm"; so clearly the spectral norm is not the default norm for all parties considered! :) As commented by @amoeba, different communities might have different terminology conventions. It goes without saying that I think Gentle's book is an invaluable resource on the matter of Lin. Algebra application in Statistics and I would prompt you to look it further!