Super　resolution　image　reconstruction　under　unknown　Gaussian　blur　has　been　a　challenging　topic.　Advanced　optimization-based　works　for　blind　image　super-resolution　(SR)　were　reported　to　be　effective,　but　there　exist　both　large　data　space　storage　and　time　consuming　due　to　vector-variable　optimization.　This　paper　proposes　a　matrix-variable　optimization　method　for　fast　blind　image　SR.　We　first　present　an　accurate　blur　kernel　estimation-based　matrix　decomposition　method.　Then　we　propose　minimizing　a　matrix-variable　optimization　problem　with　sparse　representation　and　TV　regularization　terms.　The　proposed　method　can　exactly　estimate　the　unknown　blur　kernel　and　blur　matrix.　Compared　with　vector-variable　optimization　based　methods　for　blind　image　SR,　the　proposed　method　can　greatly　reduce　their　data　space　storage　and　computation　time.　Compared　with　deep　learning　methods,　the　proposed　method　can　directly　deal　with　multiframe　SR　problem　without　training　and　learning　task.　Experimental　results　show　that　the　proposed　algorithm　is　superior　to　conventional　optimization-based　method　in　terms　of　solution　quality　and　computation　time.　Moreover,　the　proposed　method　can　obtain　higher　reconstruction　quality　than　the　deep　learning　methods,　specially　in　the　case　of　large　blur　kernels.